Joural of Mathematics Research September, 009 New Approximatios to the Mathematical Costat e Sajay Kumar Khattri Correspodig author) Stord Haugesud Uiversity College Bjørsosgate 45 PO box 558, Haugesud, Norway Tel: 47-570-83 E-mail: sajay.khattri@hsh.o Abstract Based o the Newto-Cotes ad Gaussia quadrature rules, we develop several ew closed form approximatios to the mathematical costat e. For validatig effectiveess of our approximatios, a compariso of our results to the existig approximatios is also preseted. Because of the level of mathematics, the preseted work is easily embraceable i a udergraduate class. Aother aim of this work is to ecourage studets for formulatig other better approximatios by usig the suggested strategy. Keywords: Mathematical costat, Closed form approximatio, Quadrature. Itroductio The umber e is oe of the most fudametal umbers i mathematics. This umber is also referred to as Euler s umber or Napier s costat. I this work, we develop several ew closed form approximatios to the mathematical costat e through quadrature rules. Classically the umber e is defied as = lim + ) ) Let us call this defiitio the Euler s e Ee). First year udergraduate studets are exposed to cocepts of limits ad quadrature. By usig these cocepts, we are further refiig the limit ). Based o this work, teacher ca ask studets to formulate eve better approximatios to the mathematical costat e. Figure presets a graph of the fuctio /x. The area uder the graph ad betwee the vertical lies x = ad x = + is give as + x dx For formig various closed form approximatios to e, we use quadrature rule for approximatig the itegral. The exact value of this itegral is l + ).. Approximatio through trapezoidal quadrature rule The Trapezoidal quadrature rule is give as + x dx = h [ f x ) + f x ) ] www.ccseet.org/jmr 3
Vol., No. ISSN: 9-9795 Here, h =, x = ad x = +. Thus, Thiitio of e through the trapezoidal rule is give as [l x] + = [ + ] + ) + l = + + = + 0.5 [ + 0.5 + 0.5 ] ) + l = + 0.5 ) 0.5 + + + 0.5 l = l e ) 0.5 + + + 0.5 e = ) 0.5 + + + 0.5 = lim ) Let us call this defiitio, the Trapezoidal Euler s e TEE). 3. Approximatio through Simpso s quadrature rule The Simpso s 3 quadrature rule for approximatig itegral is give as Here, h =, x 0 =, x = + ad x = + + + + x dx = h 3 x dx = ) + l = ) + 5 l + ) = + + [ ] l + ) + +5 + + = l e e = + [ ) + +5 + + [ f x0 ) + 4 f x ) + f x ) ] [ + 8 + + ] + + + 5 + + ) Thiitio of e through the Simpso s quadrature rule is = lim + [ ) + +5 + + 3) Let us call this defiitio, the 3 Simpso Euler s e 3 SEE) 4 www.ccseet.org
Joural of Mathematics Research September, 009 4. Approximatio through Simpso s 3 8 quadrature rule Approximatio of the itegral through Simpso s 3 8 quadrature rule is + x dx = 3 h 8 Here, h = 3, x 0 =, x = 3+ 3, x = 3+ 3 ad x 3 = + + x dx = l + ) = [ + [ [ f x0 ) + 3 f x ) + 3 f x ) + f x 3 ) ] 3 3 + 54 ] + 0 + 3 3 + 54 + 0 + ) + 8 + 4 + 7 8 ) + 4 + 7 + 3 3 + 54 + 0 + 8 + 4 + 7 3 3 + 54 + 0 + l + ) [+ e = + ) [+ l + ) = 8 +4 +7 3 3 +54 +0 + 8 +4 +7 3 3 +54 +0 + = l e Thiitio of e through the Simpso s 3 8 quadrature rule is = lim + ) 8 [+ +4 +7 3 3 +54 +0 + 4) Let us call this defiitio, the 3 8 Simpso Euler s e 3 8 SEE). 5. Approximatig e through Boole s quadrature rule The Boole s quadrature rule is give as follows + x dx = h 45 Here, h = 4, x 0 =, x = 4+ 4, x = 4+ 4, x 3 = 4+3 4 ad x 4 = + l + ) = 4 45 = [ 7 f x0 ) + 3 f x ) + f x ) + 3 f x 3 ) + 7 f x 4 ) ] + [ 7 + 3 4 4 + + 4 4 + + 3 4 4 + 3 + 7 ] + 480 3 + 880 + 490 + 83 90 4 + 90 3 + 0 + 0 + 7 [ 480 3 + 880 + 490 + 83 + 90 4 + 90 3 + 0 + 0 + 7 l + ) [+ e = + ) [+ l ) + ) = 480 3 +880 +490 +83 90 4 +90 3 +0 +0 +7 480 3 +880 +490 +83 90 4 +90 3 +0 +0 +7 = l e www.ccseet.org/jmr 5
Vol., No. ISSN: 9-9795 Thiitio of e through the Boole s quadrature rule is = lim + ) 480 [+ 3 +880 +490 +83 90 4 +90 3 +0 +0 +7 5) Let us call this defiitio, the Boole Euler s e BEE).. Approximatig e through Gauss-Legedre poit quadrature The two poit Gauss-Legedre Quadrature is give as + x dx = k [ w f x ) + w f x ) ] Here, k = + =, x = + + 3 ad x 3 = + 3. Weights are w = ad w =. + + + 3 l l + x dx = 3 + ) 3 + + 3 + ) 3 + ) + 3 = + + + l + ) = ) + + +3 e = e = = l e + ) + + +3 + ) 3+ [+ +3 Thiitio of e through the two poit Gauss-Legedre quadrature rule is = lim + ) 3+ [+ +3 ) Let us call this defiitio, the two poit Gauss-Legedre Euler s e P-GLEE). 7. Approximatig e through Gauss-Legedre 3 poit quadrature Three poit Gauss-Legedre quadrature rule is give as + The weights w i ad poits x i are give as x dx = k [ w f x ) + w f x ) + w 3 f x 3 ) ] w = 8 9 w = 5 9 w 3 = 5 9 x = + x = + ) 5 + 3 5 x 3 = + ) 5 3 5 www.ccseet.org
Joural of Mathematics Research September, 009 Thus, l + ) = 9 0 + 0 + + ) 5 + 5 + 3 ) 5 + 5 3 ) l + ) [ 0 ] + 0 + ) = 0 3 + 90 + 3 + 3 = 30 + 5 + 3 + 0 + 0 + + 30 + 5 + 3 0 + 0 + l + ) = l + ) [+ 30 +5+ 3 0 +0 + = l e e = ] + ) [+ 30 +5+ 3 0 +0 + Thiitio of e through the three poit Gauss-Legedre quadrature rule is = lim + ] ) [+ 30 +5+ 3 0 +0 + 7) Let us call this defiitio, the three poit Gauss-Legedre Euler s e 3P-GLEE). The Eglish meaig of the word GLEE is brightess. Through umerical work, we will see that it is ideed a very bright approximatio to the fudametal costat e. For = 00, the 3P-GLEE gives us.78888459047759, ad which is e accurate to fiftee decimal places. If we replace by as doe by Kox 999) ad Brothers 998) i their approximatio formulae, the 3P-GLEE gives exact e for = 09. 8. Approximatig e through Gauss-Legedre 4 poit quadrature The four poit Gauss-Legedre quadrature rule is give as + x dx = k [ w f x ) + w f x ) + w 3 f x 3 ) + w 4 f x 4 ) ] Here, k =. Weights w i ad poits x i are give as w = 8 + 30 3 w = 8 + 30 3 w 3 = 8 30 3 w 4 = 8 30 3 + ) 7 + 3 x = 7 + ) 7 3 x = 7 + ) 7 + 3 + x 3 = 7 + ) 7 3 x 4 = 7 5 ) 5 ) 5 ) 5 ) www.ccseet.org/jmr 7
Vol., No. ISSN: 9-9795 Thus, Thus, l + ) 40 3 + 30 + 0 + 5 = 40 4 + 840 3 + 540 + 0 + = 40 3 + 30 + 0 + 5 40 3 + 840 + 540 + 0 + = + 0 + 80 + 95 + 40 3 + 30 + 0 + 5 [ + 0 + 80 + 95 + ] l + ) = 40 3 + 30 + 0 + 5 l + ) 0 + + 80 + 95 + 40 3 + 30 + 0 + 5 = l e Thiitio of e through the four poit Gauss-Legedre quadrature rule is = lim + ] ) [+ 0 +80 +95+ 40 3 +30 +0 +5 8) Let us call this defiitio, the four poit Gauss-Legedre Euler s e 4P-GLEE). For = 00, the 4P-GLEE gives us.788884590453530 ad which is e accurate to twety oe decimal places. If we replace by as doe by Kox 999) ad Brothers 998) i their approximatio formulae, the 4P-GLEE gives exact e for = 83. 9. Approximatig e through Gauss-Legedre 5 poit quadrature The five poit Gauss-Legedre quadrature rule is give as: + Here, k =. Weights w i ad poits x i are give as x dx = k [ w f x ) + w f x ) + w 3 f x 3 ) + w 4 f x 4 ) + w 5 f x 5 ) ] w = 8 5 w = 450 + 3 900 w 3 = 450 + 3 900 w 4 = 450 + 3 900 w 5 = 450 3 900 x = + 70 x = + + 4 70 x3 = + 4 70 x4 = + + 4 70 x5 = + 4 45 4 70 45 4 70 45 + 4 70 45 + 4 70 l + ) 750 4 + 50 3 + 9870 + 30 + 37 = 750 5 + 8900 4 + 800 3 + 300 + 900 + 30 = [ 750 4 + 50 3 + 9870 ] + 30 + 37 750 4 + 8900 3 + 800 + 300 + 900 + 30 = + 3780 3 + 930 + 3990 + 73 + 30 750 4 + 50 3 + 9870 + 30 + 37 8 www.ccseet.org
Joural of Mathematics Research September, 009 Thus, [ + 3780 3 + 930 + 3990 + 73 + 30 ] 750 4 + 50 3 + 9870 + 30 + 37 l + ) = l e l + ) 3780 + 3 + 930 + 3990 + 73 + 30 750 4 + 50 3 + 9870 + 30 + 37 = l e Thiitio of e through the five poit Gauss-Legedre quadrature rule is ] = lim + ) [+ 3780 3 +930 +3990 +73+30 750 4 +50 3 +9870 +30 +37 9) Let us call this defiitio, the five poit Gauss-Legedre Euler s e 5P-GLEE). For = 00, the 5P-GLEE gives us.78888459045353087508375, ad which is e accurate to twety five decimal places. O the other had, the classical defiitio ) gives e accurate oly to two decimal places. Approximatio by various GLEE formulae ad Euler s e Ee) equatio??) for = 00. 4P-GLEE { }} { e = }{{}.7 888845904 5353 087 473549775747093999 Ee } {{ } 3P-GLEE } {{ } 5P-GLEE If we replace by as doe by Kox 999) ad Brothers 998), the 5P-GLEE gives exact e for =. After itroducig studets to the stadard defiitio of the umber e give by equatio ). Whe we preseted our ew defiitios ad their derivatios i the class, the studets has show cosiderable iterest. Studets fid it very appealig that simple techiques gives us the closed form approximatio which improves accuracy from two digits to twety five digits. 0. Numerical Work For performig computatios to high accuracy, we are usig the C ++ library ARPREC D. H. Bailey, 00). Let us ow briefly metio existig relatios for represetig mathematical costat e. Kox 999) ad Brothers 998) have also developed some very ice closed form approximatios to the mathematical costat e. The Table ) displays closed form formulae developed i Kox 999) ad Brothers 998). Reader ca observe that the formulae B, B, B3, B4, B5, B ad B7 are ot defied for =. O the other had, formulae ), 3), 4), 5), ), 7), 8) ad 9) aried for =. Let us ow compare our formulae with the formulae preseted i the Table. For = 00, Table presets error i approximatig e through differet closed form approximatios. Here, error is equal to the exact value of the mathematical costat e mius the value give by differet approximatios. From the Table, it ca be iferred that the approximatios developed by us are more accurate. It is also obvious that our formulae are computatioally efficiet. For example, i the Table the formula B7 gives most accurate approximatio. The reader ca see that for evaluatig B7, we eed to evaluate B ad B5. Ad, for computig B, we eed to compute B3 ad B4. Ad for computig B4, we eed to compute ACM ad B. Ad so o. I the differet defiitios of the costat e through formulae ), 3), 4), 5), ), 7), 8) ad 9). It ca be see that for large values of, all of these formulae behaves as: + ) +0.5 0) Let us call this defiitio, the Gauss e GE). The reader ca observe that the GE is modestly differet tha the classical defiitio ). To see why this defiitio of e is more accurate tha the classical defiitio of e. Let us compute e from these two defiitios for = 000. From the classical defiitio, we get e =,7 93 93. Which is accurate oly till 3 decimal places. While from GE we get e =,78 8 05. Ad, which is e accurate till decimal places. It is ideed a substatial improvemet over classical result. We are just chagig the classical defiitio slightly. Let us ow observe a iterestig coectio betwee the formula MIM see Table ) ad our formulae GE 0). For = 000, MIM gives e =,78 8 734. While from GE we get e =,78 8 05. Both of these values are accurate till decimal places. Thus, both of these formulae are givig same order of accuracy. This lead us to believe that they must be the same formulae. The readers are ecouraged to see it for themselves. Replacig by 0.5) i GE 0), we www.ccseet.org/jmr 9
Vol., No. ISSN: 9-9795 get MIM. These formulae are derived from differet methods. The MIM is derived i Kox 999) ad Brothers 998) by ifiite series expasio. While, we obtaied GE 0) from quadrature rules.. Coclusios We have developed several closed form approximatios to the mathematical costat e. Numerical compariso study validate the effectiveess of our results over the existig closed form approximatios. The other mai aim of this work is to ecourage udergraduate studets for developig ew approximatios. The strategy preseted i this paper is easily adoptable i a udergraduate class. Based o the work preseted i this paper, teachers ca ask studets to further develop ew approximatios by usig various other quadratures. Through our teachig experiece we foud our work is ecouragig studets to formulate eve better approximatios to the costat e. Refereces C. L. Wag. 989). Simple Iequalities Ad Old Limits. America Mathematical Mothly. Vol. 94), April. C. W. Bares. 984). Euler s costat ad e. America Mathematical Mothly. Vol. 97). D. H. Bailey, Y. Hida, X. S. Li ad B. Thompso. 00). ARPREC: A Arbitrary Precisio Computatio Package. LBNL-535. Sept. E. Maor. 994). e: The Story of a Number. Priceto Uiversity Press.. H. J. Brothers ad J. A. Kox. 998). New closed-form approximatios to the logarithmic costat e. Mathematical Itelligecer. 04). H. Yag ad H. Yag. 00). The Arithmetic-Geometric Mea Iequality ad the Costat e. Mathematics Magazie. Vol. 744) October. J. A. Kox ad H. J. Brothers. 999). Novel series based approximatios to e. The College Mathematics Joural, Vol. 30 4), September. R. Johsobaugh. 98). The Trapezoid Rule, Stirlig s Formula ad Euler s costat. America Mathematical Mothly. Vol. 889), Nov. 98. T. N. T. Goodma. 98). Maximum products ad lim + ) = e. America Mathematical Mothly. Vol. 939), October 98. 0 www.ccseet.org
Joural of Mathematics Research September, 009 Table. Differet closed form approximatios to the umber e from Kox 999) ad Brothers 998). Formulae + ) + ) + ) Name ACM MIM + ) ) 5 ) 8 + 3 + ACMMIM + ) + ) ) B ) B ) ) 8 7 B) MIM) B5 7 + ) + + ) ) B3 ) ACM) + 5 B) B4 0 7 B3) 3 B4) B 7 5 58 B) B5) B7 75 75 Table. Error exact-formulae) by differet closed form approximatios for = 00. Formulae Error Formulae Error ACM 5,07 0 5 P-GLEE,480 0 0 MIM 9,0 0 5 3P-GLEE 9,4 0 ACMMIM,507 0 4P-GLEE 5,93 0 B 3,90 0 0 5P-GLEE 3,70 0 B7,385 0 0 www.ccseet.org/jmr
Vol., No. ISSN: 9-9795 Figure. Graph of f x) = x. The shaded area is equal to l + ). www.ccseet.org