Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

 Hortense Cook
 3 months ago
 Views:
Transcription
1 Applied Mathematical Scieces, Vol. 7, 03, o. 6, HIKARI Ltd, Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series Nichaphat Pataarapeelert, ad Vimolyut Varasavag Departmet of Mathematics, Faculty of Applied Sciece Kig Mogkut s Uiversity of Techology North Bagkok 8 Pracharat Sai Road, Wogsawag, Bagsue Bagkok 0800, Thailad Copyright 03 Nichaphat Pataarapeelert ad Vimolyut Varasavag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Chebyshev ad Legedre series have bee widely used i may areas of physics ad egieerig. I this study, we eamie o the efficacy of both seris i terms of series approimatio ad covergece. Some eamples of fuctios that ca be epaded i the form of ifiite series of Chebyshev ad Legedre polyomials are cosidered. Usig the same umber of polyomials coefficiets, we fid that Chebyshev series give a better approimatio tha Legedre series. I additio, we fid that the rate of covergece of ifiite series geerated from the same fuctio i terms of Chebyshev polyomial are more rapid tha Legedre polyomial for while for, Chebyshev ad Legedre series give the close rate of covergece. Fially, the covergece of Chebyshev ad Legedre series with the same coefficiets are cosidered. Keywords: Chebyshev polyomial, Legedre polyomial, Series, Covergece
2 36 Nichaphat Pataarapeelert ad Vimolyut Varasavag Itroductio Special fuctios arise i may problems of pure ad applied mathematics, mathematical statistics, physics, ad egieerig. I particular, they play a importat role i umerical aalysis as see i approimatio of itegrals ad solutio of differetial equatios i physic problems e.g. electricity problems [3], [6]. Some special fuctios are ofte employed i approimatig a give fuctio as a appropriate polyomial. Especially, polyomials cosiderig with low order are easy to hadle ad ca be performed with miimal pealty i accuracy for computatioal speed [7]. I additio to use trucated Taylor epasio, Chebyshev ad Legedre polyomials are also helpful tools to approimate cotiuous fuctios [3]. However, the compariso of which series gives better approimatio may ot be straightforward. I this research, we fill this gap by rearragig Chebyshev ad Legedre series i the form of power series so that we are able to compare the accuracy of approimatio. We will give some eamples of epadig fuctios i term of ifiite series of Chebyshev Polyomials ad Legedre Polyomials. Usig these epasios, we will approimate the fuctios uder the same umber of polyomials coefficiets. Moreover, we compare the rate of covergece of ifiite series i terms of Chebyshev ad Legedre polyomials that are epaded from the same fuctio. Fially, we aalyze ad determie the coditio for covergece of ifiite series represeted i terms of Chebyshev or Legedre polyomials. Usig such coditios, we compare the rate of covergece of series of Chebyshev ad Legedre polyomials where the coefficiets of such series are the same. Approimatios of Fuctios Usig Chebyshev ad Legedre Series Epasios Cosider a fuctio f ( ) where that ca be writte i the form of Chebyshev series epasio as [6] f( ) ct( ) () 0 where T ( ) is Chebyshev polyomial of the first kid of degree. The coefficiets ad c determied from the orthogoal property are give by f( T ) ( ) c 0 d ()
3 Compariso study of series approimatio 37 f( T ) ( ) c ;,, 3,... d (3) For Legedre series epasio of f, ( ) the series is i the followig form where ( ) coefficiet a is give by f ( ) ap ( ) (4) 0 P is Legedre polyomial of the first kid of degree ad the a f ( ) P ( ) d ; 0,,, 3,.... () As kow that the fiite Chebyshev ad Legedre series epasios are very useful i the approimatio of various fuctios, here we preset some fuctios with their Chebyshev ad Legedre series epasio forms ad the approimate their partial sums as follows.. Trigoometric Fuctio We first epad a fuctio f ( ) si ;, (6) as Chebyshev series ct ( ) 0. To determie the coefficiets c, we substitute cos i equatios (3). We the have 4 c ;, 3,,.... (7) Sice si is odd fuctio, we fid that c 0 for 0,, 4, 6,.... Therefore, Chebyshev series epasio of si is i the form 4 si ( ) m m 0 (m ) T. (8) To approimate this series, we will cosider its partial sum for five terms as 4 si T ( ) T ( ) T ( ) T ( ) T ( ). (9) we substitute T ( ), T 3 () 4 3, T 3 ( ) 6 0, T ( ) , ad T ( ) i equatio (9). Therefore, the iverse sie fuctio ca be writte i term of degree of with a five terms trucated series as si (0)
4 38 Nichaphat Pataarapeelert ad Vimolyut Varasavag Further, we will fid Legedre series epasio ap ( ) of (6). 0 Employig orthogoal relatio, we fid that that a 0for, 4, 6,... while.3...( ) a ad a ( ( ) ( )) ; 3,, 7,... P P ( ) () We hece obtai Legedre series epasio as.3...(m ) f ( ) P ( ) ( ( ) ( )) P P m m m ( m). () Similarly, usig the same umber of polyomials coefficiets as Chebyshev series 3 epasio, ad substitutig P( ), P ( ) ( 3 ), P ( ) (63 70 ), P 7( ) ( ), ad P9 ( ) ( ), we have si (3) The comparisos of usig these two approimatios are show i followig Table ad Table. Table. Approimatios of f ( ) si obtaied from equatios (0) ad (3) by usig Chebyshev ad Legedre polyomials, respectively. si Chebyshev series Legedre series epasio epasio Table. Errors compariso from calculatig five terms of series epasios betwee Chebyshev ad Legedre of f ( ) si. Error from usig Error from usig Chebyshev series epasio Legedre series epasio
5 Compariso study of series approimatio 39 From Table, we fid that the values from Chebyshev series epasio for five terms are more accurate tha Legedre series epasio whe the umber of polyomials coefficiets is the same. However, we fid that whe, the error of usig Chebyshev is while it is zero from Legedre. Moreover, whe 0, the error from usig partial sums of both of Chebyshev ad Legedre have is zero.. Epoetial Fuctio For epoetial fuctio, f ( ) e ;, (4) it ca be approimated by usig Chebyshev ad Legedre series epasios as the previous fuctio. To compare the efficiecy of usig these series, we proceed as follows. We begi with determiig the coefficiets of Chebyshev series epasio as give i ()(3). Substitutig cos, we fid that cos c0 e d () 0 ad cos c e cos d ;,, 3,... (6) 0 We calculate the itegrals umerically i equatios () ad (6) by employig Simpso s 3 rule with the equal space h. We the obtai 0 h cosh cos h cos9h c0 e 4e e... 4e e.66 3, (7) h cosh cos h cos9h c e 4e cosh e cos h... 4e cos9h e (8) We ivestigate c, c3, c 4 by usig similar process ad the obtai c 0.7, c , adc Therefore, the approimatio of epoetial fuctio i term of partial sum of Chebyshev polyomial is give by e.66 T ( ).303 T ( ) 0.7 T ( ) T ( )+0.00 T ( ) (9) After substitutig T ( ) for 0,,, 3, 4 i terms of, we obtai 3 4 e (0) For the coefficiets of Legedre series epasio give i equatio (), we fid that a e P( ) d ; 0,,, 3,.... ()
6 330 Nichaphat Pataarapeelert ad Vimolyut Varasavag We ca determie a for 0,,, 3, 4, by substitutig P ( ) i term of ad usig by part itegratio. Fially, we obtai the approimatio of epoetial fuctio writte i the form of partial sum of Legedre polyomials 4 0 ap ( ) as e.7 P0( ).036 P( ) P( ) P3( )+0.0 P4( ). () We ote that this is the same umber of polyomials coefficiets as Chebyshev series i (9). Agai, we rewrite () i term of as 3 4 e (3) The comparisos of usig these two approimatios are show as followig Table 3 ad Table 4. Table 3. Approimatios of f ( ) e obtaied from equatios (0) ad (3) by usig Chebyshev ad Legedre polyomials, respectively. e Chebyshev series Legedre series epasio epasio Table 4. Errors compariso from calculatig five terms of series epasios betwee Chebyshev ad Legedre of f ( ) e. Error from usig Chebyshev series epasio Error from usig Legedre series epasio From Table 4, we observe that the errors from usig Chebyshev series epasio of f ( ) e are smaller tha Legedre series epasio whe we calculate the partial sum for five terms. For this five terms calculatio, however, we ote that at 0 the error obtaied from usig Chebyshev is while it is zero whe we use Legedre series epasio.
7 Compariso study of series approimatio 33 3 The Rate of Covergece of Chebyshev ad Legedre Series Epasios 3. The series epaded from the same fuctio It is kow that both of T ( ) ad P ( ) are orthogoal polyomials for. This meas that if we defie some aalytic fuctios for, we may write those fuctios i term of T ( ) ad P ( ) give by () ad (4), respectively. Although, both series coverge, the rate of covergece may be differet. I this sectio, we study the rate of covergece of series obtaied from Chebyshev ad Legedre epasios. For, we cosider the step fuctio give by 0 ; 0 f ( ) (4) ; 0. This give fuctio ca be writte i the form of Chebyshev ad Legedre series epasios as m f ( ) ( ) T ( 0 ) ( ) ( ) m T m 4m T, () ad m ( ) (m )!(4m ) f ( ) P( ) P ( ) ( ) 0 P, (6) m 4 m m ( m )!( m )! respectively. To compare the rate of covergece betwee these two series, we determie the value of ide m used for umerical calculatio for each series uder the tolerace of order 0. The results are show i Table as follows. Table. Value of ide m for covergece of () ad (6) subject to the error tolerace of 0. m for Chebyshev m for Legedre We observe from Table that the ifiite series of Chebyshev i () requires less term for approimatio tha ifiite series of Legedre i (6) uder the tolerace of order 0. Therefore, we ca coclude that Chebyshev series epasio are quite coverge rapidly with respect to Legedre series epadig from the step fuctio (4).
8 33 Nichaphat Pataarapeelert ad Vimolyut Varasavag I a iterestig case, we focus o the covergece of ifiite series i term of Chebyshev ad Legedre polyomials for. I the followig eample, we cosider the Chebyshev ad Legedre series epasio of the fuctio l. Let 8 8 w 7 8 (7) 8 where w, we fid that. Sice w 87w l l 8 l 8 7w 8 7w (8) ad this fuctio is odd, we ca fid the series epasio of l as 8 7w 8 l c T ( w) m m 8 7w (9) m0 where cos c l cos(m ) d m cos (30) ad 8 7w 8 l a P ( w) m m 8 7w (3) 0 where 4m 3 87w a 8 l P ( w) dw m m 8 7w (3) which are represeted i terms of Chebyshev ad Legedre polyomials, respectively. We et determie the value of ide m used for umerical calculatio for each series for l uder the tolerace of order 0. The results are show i Table 6. Table 6. Value of ide m for covergece of (9) ad (3) subject to the error tolerace of 0. m for Chebyshev m for Legedre We observe from Table 6 that the ifiite series of Chebyshev i (9) requires more or equal terms i approimatio tha Legedre i (3). Therefore, we might
9 Compariso study of series approimatio 333 coclude that the rate of covergece of Legedre series epaded from l is close to Chebyshev series epasio i this case. 3. The series with the same coefficiet I this sectio, we fid the coditios for covergece of Chebyshev series ct ( ) ad Legedre series 0 ap ( ). For, the ifiite 0 series i term of Chebyshev polyomial is absolutely coverget if ct ( ) 0 coverges. Sice T ( ) cos, the ecessary coditio for absolute covergece is that c must coverge. While, the ifiite series i term of 0 Legedre polyomial is absolutely coverget if ap coverges. 0 Sice P ( ), this implies that the ecessary coditio is that coverge. ad I additio, we determie the coditios for covergece of 0 a must 0 0 ct ( ) ap ( ) whe. For Chebyshev series, we let cosh t ; t 0. By usig the geeratig fuctio for ( ) Therefore, the Chebyshev series T, we fid that T (cosh t) cosh t. ct ( ) for is i the form f ( ) c T (cosh t) c cosh t. (33) Usig the ratio test of covergece, we get c cosh( ) t lim c cosh t Lettig L lim Moreover, sice c e e lim t t c e e c t lim e. c c c t e ( ) t ( ) t, the Chebyshev series is absolutely coverget if e t e t (34). (3) L, the coditio (3) becomes
10 334 Nichaphat Pataarapeelert ad Vimolyut Varasavag L L. (36) We et cosider the ifiite series preseted i term of Legedre polyomials for. Similarly, we defie cosh t ; t 0. The closed form approimatio of Legedre polyomial P ( ) for large [8] is i the form ( ) t e P (cosh t). (37) siht Therefore, the ifiite series i terms of Legedre polyomial the form ap ( ) is i 0 ( ) t N e ap(cosh t) (cosh ) ap t a. (38) 0 0 N siht whe N is sufficietly large. This series coverges whe the secod term o the right side of (38) coverges. Usig the same procedure as Chebyshev series, the ecessary coditio for absolutely covergece of ifiite series i terms of Legedre series is give by [8] M M (39) a where lim M. We observe that the covergece of such series deped a o the behavior of their coefficiets. We ow compare the rate of covergece of ifiite series i terms of Chebyshev ad Legedre polyomials where the coefficiets for both series are the same. For, employig the give coefficiet c a, (40) ( ) we fid that c a coverges which is satisfied the 0 0 0( ) above coditio. Therefore, the ifiite series of Chebyshev ad Legedre polyomials are absolutely coverget. To compare the rate of covergece, we determie the umber of terms for each series uder the tolerace of order 0. After calculatig, we fid that the ifiite series i terms of Chebyshev polyomials usig the same coefficiet employ much more term to coverge tha Legedre polyomials as show i Table 7.
11 Compariso study of series approimatio 33 Table 7. The umber of term for covergece of T ( ) ad 0 ( ) P ( ) subject to the error tolerace of 0. ( ) 0 Number of term for Number of term for Chebyshev Legedre For, we cosider the give coefficiet c a 0.. (4) We fid from (4) that L M 0.. Therefore, the rage of satisfied with the coditios (36) ad (39) that is.0. Here, the umbers of terms used for covergece of these series are show i Table 8. Table 8. The umber of term for covergece of subject to the error tolerace of 0. Number of term for Chebyshev 0 0. T ( ) ad 0. P ( ) 0 Number of term for Legedre We observe that ifiite series of Chebyshev polyomials requires more terms, uder the same coefficiet, for covergece tha Legedre polyomials for both case ad. 4 Coclusio I summary, the elemetary fuctios were eemplified i order to compare the accuracy of approimatio i which both series are rearraged i the form of trucated power series of five terms. Although, it is observed that Chebyshev series gives a better approimatio, the test for more geeral case which eeds theoretical support is a key issue for further study. Nevertheless, our results may be useful i practical ad ca be used as a guide for deeper ivestigatio.
12 336 Nichaphat Pataarapeelert ad Vimolyut Varasavag I additio, we compared the rate of coverget of Chebyshev ad Legedre series epasio epaded from the same fuctio. Whe, the result showed that Chebyshev series epasio of the give fuctio is quite covergig rapidly with respect to Legedre series epasio. While whe, the result showed that the rate of covergece of Legedre series is similar to Chebyshev series. Fially, we provide some ifiite series i terms of Chebyshev ad Legedre polyomials uder the same coefficiet. We foud that the series i term of Chebyshev requires more terms for covergece tha Legedre polyomials for both case ad. Although, these eperimets might ot be claimed for compariso of the rate of covergece from usig Chebyshev ad Legedre polyomials because they both coverge to the differet fuctios, oe may study further about the characteristic of those fuctios ad ivestigate why Legedre polyomials require less umber of terms tha Chebyshev polyomials uder the same coefficiet. Ackowledgemet This research is supported by Sciece ad Techology Research Istitute, Kig Mogkut s Uiversity of Techology North Bagkok. Refereces [] A. Gil, J. Segura, ad N. M. Temme, Numerical Methods for Special Fuctios, Society for Idustrial ad Applied Mathematics, 007. [] G. E. Adrews, R. Askey ad R. Roy, Special Fuctios, Cambridge Uiversity Press, 000. [3] H. Wag ad S. Xiag, O the covergece rates of Legedre approimatio, Mathematics of Computatio, 8(0), [4] J. Maso, Chebyshev Polyomials: Theory ad Applicatios. Kluwer Academic, 996. [] J. C. Maso ad D. C. Hadscomb, Chebyshev Polyomials, CRC Press, New York, 003. [6] M. A. Abutheraa ad D. Lester, Computable fuctio represetatios usig effective Chebyshev polyomial, World Academy of Sciece, Egieerig ad Techology, (3)007,
13 Compariso study of series approimatio 337 [7] M. A. Cohe ad C. O. Ta, A polyomial approimatio for arbitrary fuctios, Applied Mathematics Letters, 0. [8] N. Bookorkuea, V. Varasavag, ad S. Rataapu, A Effective Method for Calculatig the Sum of a Ifiite Series of Legedre Polyomails, 8(00), 3. [9] R. E. Attar, Legedre Polyomials ad Fuctios, Createspace, 009. [0] Z. C. She ad J. Jiamig, Computatio of Special Fuctios, WileyItersciece, New York, 996. Received: April, 03
62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSolving a Nonlinear Equation Using a New TwoStep Derivative Free Iterative Methods
Applied ad Computatioal Mathematics 07; 6(6): 384 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 385605 (Prit); ISSN: 38563 (Olie) Solvig a Noliear Equatio Usig a New TwoStep
More informationf x x c x c x c... x c...
CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate
More informationAre the following series absolutely convergent? n=1. n 3. n=1 n. ( 1) n. n=1 n=1
Absolute covergece Defiitio A series P a is called absolutely coverget if the series of absolute values P a is coverget. If the terms of the series a are positive, absolute covergece is the same as covergece.
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationExample 2. Find the upper bound for the remainder for the approximation from Example 1.
Lesso 8 Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the CauchyRiema equatios? Solutio. a) A fuctio f : G C is called
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationSection 11.6 Absolute and Conditional Convergence, Root and Ratio Tests
Sectio.6 Absolute ad Coditioal Covergece, Root ad Ratio Tests I this chapter we have see several examples of covergece tests that oly apply to series whose terms are oegative. I this sectio, we will lear
More informationTopic 9  Taylor and MacLaurin Series
Topic 9  Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationTwostep Extrapolated Newton s Method with High Efficiency Index
Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 Twostep Etrapolated Newto s Method with High Efficiecy Ide V.B. Kumar Vatti Dept. of Egieerig Mathematics Adhra Uiversity Visakhapatam Idia.
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More information5.6 Absolute Convergence and The Ratio and Root Tests
5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces
More informationOn the Weak Localization Principle of the Eigenfunction Expansions of the LaplaceBeltrami Operator by Riesz Method ABSTRACT 1.
Malaysia Joural of Mathematical Scieces 9(): 337348 (05) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Joural homepage: http://eispemupmedumy/joural O the Weak Localizatio Priciple of the Eigefuctio Expasios
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationSolutions to Tutorial 5 (Week 6)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationTwo Topics in Number Theory: Sum of Divisors of the Factorial and a Formula for Primes
Iteratioal Mathematical Forum, Vol. 2, 207, o. 9, 929935 HIKARI Ltd, www.mhiari.com https://doi.org/0.2988/imf.207.7088 Two Topics i Number Theory: Sum of Divisors of the Factorial ad a Formula for Primes
More informationJANE PROFESSOR WW Prob Lib1 Summer 2000
JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or
More informationFourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oedimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
More informationTeaching Mathematics Concepts via Computer Algebra Systems
Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) EISSN: 4767 PISSN:  4759 Volume 4 Issue 7 September. 6 PP Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,
More informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
More informationConvergence of Binomial to Normal: Multiple Proofs
Iteratioal Mathematical Forum, Vol. 1, 017, o. 9, 399411 HIKARI Ltd, www.mhikari.com https://doi.org/10.1988/imf.017.7118 Covergece of Biomial to Normal: Multiple Proofs Subhash Bagui 1 ad K. L. Mehra
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationThe Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].
The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More informationA generalization of Morley s congruence
Liu et al. Advaces i Differece Euatios 05 05:54 DOI 0.86/s3660505686 R E S E A R C H Ope Access A geeralizatio of Morley s cogruece Jiaxi Liu,HaoPa ad Yog Zhag 3* * Correspodece: yogzhag98@63.com 3
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationStability Analysis of the Euler Discretization for SIR Epidemic Model
Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we
More information11.6 Absolute Convergence and the Ratio and Root Tests
.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does
More informationTHE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes
THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates.
More informationQBINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 QBINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsloly.et Received:
More informationA RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider ksample ad chage poit problems for idepedet data i a
More informationLøsningsførslag i 4M
Norges tekisk aturviteskapelige uiversitet Istitutt for matematiske fag Side 1 av 6 Løsigsførslag i 4M Oppgave 1 a) A sketch of the graph of the give f o the iterval [ 3, 3) is as follows: The Fourier
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationBeyond simple iteration of a single function, or even a finite sequence of functions, results
A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are
More informationSolutions to Homework 1
Solutios to Homework MATH 36. Describe geometrically the sets of poits z i the complex plae defied by the followig relatios /z = z () Re(az + b) >, where a, b (2) Im(z) = c, with c (3) () = = z z = z 2.
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More informationA Simplified Binet Formula for kgeneralized Fibonacci Numbers
A Simplified Biet Formula for kgeeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationOn Infinite Series Involving Fibonacci Numbers
Iteratioal Joural of Cotemporary Mathematical Scieces Vol. 10, 015, o. 8, 363379 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.1988/ijcms.015.594 O Ifiite Series Ivolvig Fiboacci Numbers Robert Frotczak
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537545 ISSN: 13118080 (prited versio); ISSN: 13143395 (olie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationAbstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial()); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationGamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 45646, USA cheg@cs.uky.edu Abstract
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationIf we want to add up the area of four rectangles, we could find the area of each rectangle and then write this sum symbolically as:
Sigma Notatio: If we wat to add up the area of four rectagles, we could fid the area of each rectagle ad the write this sum symbolically as: Sum A A A A Liewise, the sum of the areas of te triagles could
More information(p, q)baskakovkantorovich Operators
Appl Math If Sci, No 4, 55556 6 55 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://ddoiorg/8576/amis/433 p, qbasaovkatorovich Operators Vijay Gupta Departmet of Mathematics, Netaji
More informationFrequency Response of FIR Filters
EEL335: DiscreteTime Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steadystate
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationThe Choquet Integral with Respect to FuzzyValued Set Functions
The Choquet Itegral with Respect to FuzzyValued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to realvalued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationlecture 3: Interpolation Error Bounds
6 lecture 3: Iterpolatio Error Bouds.6 Covergece Theory for Polyomial Iterpolatio Iterpolatio ca be used to geerate lowdegree polyomials that approimate a complicated fuctio over the iterval [a, b]. Oe
More informationRearranging the Alternating Harmonic Series
Rearragig the Alteratig Harmoic Series Da Teague C School of Sciece ad Mathematics teague@cssm.edu 00 TCM Coferece CSSM, Durham, C Regroupig Ifiite Sums We kow that the Taylor series for l( x + ) is x
More informationRepresenting Functions as Power Series. 3 n ...
Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σalgebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationMath 140A Elementary Analysis Homework Questions 31
Math 0A Elemetary Aalysis Homework Questios .9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are ozero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationDe la Vallée Poussin Summability, the Combinatorial Sum 2n 1
J o u r a l of Mathematics ad Applicatios JMA No 40, pp 520 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationThe ArakawaKaneko Zeta Function
The ArakawaKaeko Zeta Fuctio MarcAtoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F0608 Nice Cedex 2 FRANCE MarcAtoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr
More informationOrthogonal polynomials derived from the tridiagonal representation approach
Orthogoal polyomials derived from the tridiagoal represetatio approach A. D. Alhaidari Saudi Ceter for Theoretical Physics, P.O. Box 374, Jeddah 438, Saudi Arabia Abstract: The tridiagoal represetatio
More information0.1. Geometric Series Formula. This is in your book, but I thought it might be helpful to include here. If you have a geometric series
Covergece tests These otes discuss a umer of tests for determiig whether a series coverges or 0.. Geometric Series Formula. This is i your oo, ut I thought it might e helpful to iclude here. If you have
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationAbout the use of a result of Professor Alexandru Lupaş to obtain some properties in the theory of the number e 1
Geeral Mathematics Vol. 5, No. 2007), 75 80 About the use of a result of Professor Alexadru Lupaş to obtai some properties i the theory of the umber e Adrei Verescu Dedicated to Professor Alexadru Lupaş
More information5 Sequences and Series
Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the th term The sequece {a, a 2, a 3,..., a,..., } is a
More informationa 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n
60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationThe Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs
Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology,
More informationMath 163 REVIEW EXAM 3: SOLUTIONS
Math 63 REVIEW EXAM 3: SOLUTIONS These otes do ot iclude solutios to the Cocept Check o p8. They also do t cotai complete solutios to the TrueFalse problems o those pages. Please go over these problems
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? 1. My Motivation Some Sort of an Introduction
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I taught Topological Groups at the Göttige Georg August Uiversity. This
More informationReal Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim
Real Aalysis Fall 004 Take Home Test SOLUTIONS. Use the defiitio of a limit to show that (a) lim si = 0 (b) Proof. Let ε > 0 be give. Defie N >, where N is a positive iteger. The for ε > N, si 0 < si
More informationBINOMIAL COEFFICIENT AND THE GAUSSIAN
BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,
More informationTHREE CURIOUS DIOPHANTINE PROBLEMS
JP Joural of Mathematical Scieces Volume 8, Issues &, 0, Pages 0 0 Ishaa Publishig House This paper is available olie at http://www.iphsci.com THREE CRIOS DIOPHATIE PROBLEMS A. VIJAYASAKAR, M. A. GOPALA
More information