THE TEACHING OF MATHEMATICS 009 Vol XII pp 7 4 NEW CLOSE FORM APPROXIMATIONS OF l + x) Sajay Kumar Khattri Abstract Based o Newto-Cotes ad Gaussia quadrature rules we develop several closed form approximatios to l + x) We also compare our formulae to the Taylor series expasio Aother objective of our work is to ispire studets to formulate other better approximatios by usig the preseted approach Because of the level of mathematics the preseted work is easily embraceable i a udergraduate class ZDM Subject Classificatio: N4; AMS Subject Classificatio: 00A3 Key words ad phrases: Quadrature rules; closed form approximatio; logarithm Itroductio The expressio l + x) is a importat expressio i mathematics It shows up surprisigly at may places [ 4 Geerally we approximate l + x) through a fiite sum of a ifiite series [3 But the umber of terms i a fiite sum ca make the algebra very complicated Thus we develop some simple but robust closed form approximatios to l + x) through quadrature rules The Taylor series expasio of l + x) is ) l + x) = ) i= i xi i for < x First year udergraduate studets are exposed to cocepts of limits ad quadrature Through these cocepts we develop closed form approximatios The work preseted i this paper is crucial for ehacig cocepts such as covergece ad approximatio of udergraduate studets The work will ecourage studets to formulate other relatios for represetig l + x) by usig other quadrature rules for example Lobatto [9 Based o our work teacher ca ask studets to formulate eve better approximatios to the mathematical expressio l + x) 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 The exact value of this itegral is l + ) For formig various closed form approximatios we will approximate this itegral through differet quadrature rules
8 S K Khattri Fig Approximatio through trapezoidal quadrature rule Trapezoidal quadrature rule [4 is give as + Here h = x = ad x = + Thus x dx h [fx ) + fx ) [l x + [ + + l + ) [ + = [ + 0 + + Now replacig by x i the above equatio we get the expressio of l + x) through the trapezoidal rule ) l + x) x [ + 0 x + x Let us call this expressio the Trapezoidal Euler s Log TELOG)
New close form approximatios of l + x) 9 3 Approximatio through Simpso s quadrature rule The Simpso s 3-quadrature rule [4 for approximatig itegrals is give as + x dx h 3 [fx 0) + 4fx ) + fx ) Here h = x 0 = x = + ad x = + + x dx [ + 8 + + + l + ) [ 8 + 3) + + + Now replacig by x i the above equatio we get the expressio for l + x) through the Simpso s quadrature rule as 4) l + x) [ x + 8x + x + x + x Let us call this expressio the 3 -Simpso Euler s Log 3 -SELOG) 4 Aproximatio through Simpso s 3 8-quadrature rule Approximatio of the itegral through Simpso s 3 8-quadrature rule [4 is + x dx 3h 8 [fx 0) + 3fx ) + 3fx ) + fx 3 ) Here h = 3 x 0 = x = 3+ 3 x = 3+ 3 ad x 3 = + + x dx [ 3 3 + 4 + 0 + 3 3 + 4 + 0 + ) + 8 + 4 + 7 l + ) [ 9 9 8 + ) 3 + + 3 + + + Now replacig by x i the above equatio we get the expressio of l + x) through the Simpso s quadrature rule ) l + x) [ x + 9x 8 3 + x + 9x 3 + x + x + x Let us call this expressio the 3 8 -Simpso Euler s Log 3 8 -SELOG) Approximatio through Boole s quadrature rule The Boole s quadrature rule [3 is + h dx x 4 [7fx 0) + 3fx ) + fx ) + 3fx 3 ) + 7fx 4 )
0 S K Khattri Here h = 4 x 0 = x = 4+ 4 x = 4+ 4 x 3 = 4+3 4 ad x 4 = + l + ) [ 7 4 4 + 3 4 4 + + 4 4 + + 3 4 4 + 3 + 7 + Now replacig by x i the above equatio we get the expressio of l + x) through the Boole s rule 7) l + x) [ 7x + 8x 90 4 + x + 48x 4 + x + 8x 4 + 3x + 7x + x Let us call this expressio the Boole Euler s Log BELOG) Approximatio through Gauss-Legedre poit quadrature The two poit Gauss-Legedre Quadrature [ is + x dx k [w fx ) + w fx ) Here k = + = x = + + ad x 3 3 = + Weights are w 3 = ad w = [ + x dx 3 + ) 3 + + 3 + ) 3 l + ) + 3 + + = + 3 + + Now replacig by x i the above equatio we get the expressio for l + x) through the two poit Gauss-Legedre quadrature rule as 8) l + x) x + 3x + x + x Let us call this expressio the two poit Gauss-Legedre Log P-GLLOG) 7 Approximatio 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 fx ) + w fx ) + w 3 fx 3 ) w = 8 9 x = + w = 9 x = + ) + 3 w 3 = 9 x 3 = + ) 3
New close form approximatios of l + x) Thus l + ) [ 0 + 0 + 9 + ) + + 3 ) + 3 ) [ 0 [ 0 + 0 + = 0 3 + 90 = + 0 + 3 + 3 + 3 0 + 90 + 3 + 3 3 Now replacig by x i the above equatio we get the expressio l + x) through the three poit Gauss-Legedre quadrature rule as [ 0x + 0x + x 3 9) l + x) 0 + 90x + 3x + 3x 3 Let us call this defiitio the three poit Gauss-Legedre Log 3P-GLLOG) 8 Approximatio through Gauss-Legedre 4 poit quadrature The four poit Gauss-Legedre quadrature rule [ is give as + x dx k [w fx ) + w fx ) + w 3 fx 3 ) + w 4 fx 4 ) Here k = Weights w i ad poits x i are give as w = 8 + + ) 7 + 3 30 x = 3 7 Thus w = 8 + + ) 7 3 30 x = 3 7 w 3 = 8 + ) 7 + 3 + 30 x 3 = 3 7 w 4 = 8 + ) 7 30 x 4 = 3 7 3 l + ) 40 3 + 30 + 0 + 40 4 + 840 3 + 40 + 0 + 40 = + 30 + 0 + 3 4 40 + 840 + 40 + 0 + 3 4 Now replace by x i the above equatio The expressio for l + x) through the four poit Gauss-Legedre quadrature rule is 0) l + x) 40x + 30x + 0x 3 + x 4 40 + 840x + 40x + 0x 3 + x 4 Let us call this expressio the four poit Gauss-Legedre Log 4P-GLLOG)
S K Khattri 9 Approximatio through Gauss-Legedre poit quadrature The five poit Gauss-Legedre quadrature rule [ is give as + x dx k [w fx ) + w fx ) + w 3 fx 3 ) + w 4 fx 4 ) + w fx ) Here k = Weights w i ad poits x i are give as w = 8 x = + w = 40 + 3 900 70 x = + + 4 w 3 = 40 + 3 900 70 x3 = + 4 w 4 = 40 + 3 900 70 x4 = + + 4 w = 40 3 900 70 x = + 4 4 4 70 4 4 70 4 + 4 70 4 + 4 70 l + ) 70 4 + 0 3 + 9870 + 30 + 37 70 + 8900 4 + 800 3 + 300 + 900 + 30 = 70 + 0 + 9870 + 30 3 + 37 4 70 + 8900 + 800 + 300 + 900 3 + 30 4 Now replacig by x i the above equatio the expressio for l + x) through the five poit Gauss-Legedre quadrature rule is obtaied ) l + x) 70x + 0x + 9870x 3 + 30x 4 + 37x 70 + 8900x + 800x + 300x 3 + 900x 4 + 30x Let us call this expressio the five poit Gauss-Legedre Log P-GLLOG) 0 Numerical work For performig computatios to great accuracy we use the high precisio C ++ library ARPREC [4 I almost every calculus book the mathematical expressio l + x) is give by the followig ifiite series: ) l + x) = i= i xi ) i = x x + x3 3 x4 4 x + Let us ow fid out the error i computig l by various of our formulae ad ifiite series For exact value of l we are usig the library ARPREC For
New close form approximatios of l + x) 3 computig l by the ifiite series we are cosiderig first 0 terms Table presets error i approximatig l through differet closed form approximatios Here error is equal to the exact value of the mathematical costat l mius the value give by differet approximatios From Table it ca be iferred that our approximatios are more accurate Formulae Ifiite series Error 48 774 0 3-SELOG 97 4 0 3 P-GLLOG 8394 883 0 4 3P-GLLOG 4P-GLLOG P-GLLOG 48 744 0 73 4 0 7 70 9 0 8 Table Error exact-formulae) by differet closed form approximatios We are takig first te terms of the ifiite series Coclusios I this work we have developed some ew closed form approximatios for the expressio l + x) Numerical work autheticates the robustess of these closed form approximatios Oe big advatage of the formulae over series is that the formulae ca be easily programmed eve o a calculator Basic mathematics is beig used to derive these relatios Thus the preseted strategy is easily adopted i a udergraduate class It will ecourage studets i formulatig eve more improved formulae for l + x) Ackowledgmets We thak reviewers for their time ad for their valuable commets REFERENCES J A Kox ad H J Brothers Novel series based approximatios to e The College Mathematics Joural 30 4) 999 E Maor e: The Story of a Number Priceto Uiversity Press 994 3 H J Brothers ad J A Kox New closed-form approximatios to the logarithmic costat e Mathematical Itelligecer 0 4) 998 4 D H Bailey Y Hida X S Li ad B Thompso ARPREC: A Arbitrary Precisio Computatio Package LBNL-3 Available http://wwwostigov/bridge/servlets/purl/8734- szjom/ative/8734pdf Accessed 0 August 008 T N T Goodma Maximum products ad lim + ) = e America Mathematical Mothly 93 9) 98 C L Wag Simple iequalities ad old limits America Mathematical Mothly 9 4) 989 7 H Yag ad H Yag The Arithmetic-Geometric Mea Iequality ad the costat e Mathematics Magazie 74 4) 00
4 S K Khattri 8 C W Bares Euler s costat ad e America Mathematical Mothly 9 7) 984 9 S Wasowicz O error bouds for Gauss-Legedre ad Lobatto Quadrature Rules J Iequalities Pure Appl Math 7 3): Article 84 http://wwwemisde/jourals/jipam/images/004 0 JIPAM/004 0 wwwpdf Accessed 30 August 008 0 R Johsobaugh The Trapezoid Rule Stirlig s Formula ad Euler s costat America Mathematical Mothly 88 9) 98 E W Weisstei Lobatto Quadrature MathWorld A WolframWeb Resource http://mathworldwolframcom/lobattoquadraturehtml Accessed 30 August 008 E W Weisstei Legedre-Gauss Quadrature MathWorld A Wolfram Web Resource http://mathworldwolframcom/legedre-gaussquadraturehtml Accessed 0 February 009 3 E W Weisstei Boole s Rule MathWorld A Wolfram Web Resource http://mathworldwolframcom/boolesrulehtml Accessed 0 February 009 4 E W Weisstei Newto-Cotes Formulas MathWorld A Wolfram Web Resource http://mathworldwolframcom/newto-cotesformulashtml Accessed 0 February 009 Stord/Haugesud Uiversity College Bjørsosgt 4 Haugesud 8 Norway E-mail: sajaykhattri@hsho