THE TEACHING OF MATHEMATICS 2009, Vol. XII, 1, pp. 7–14 NEW CLOSE FORM APPROXIMATIONS OF ln(1 + x) Sanjay Kumar Khattri Abstract. Based on Newton-Cotes and Gaussian quadrature rules, we develop several closed form approximations to ln (1 + x). We also compare our formulae to the Taylor series expansion. Another objective of our work is to inspire students to formulate other better approximations by using the presented approach. Because of the level of mathematics, the presented work is easily embraceable in an undergraduate class. ZDM Subject Classification: N45; AMS Subject Classification: 00A35. Key words and phrases: Quadrature rules; closed form approximation; logarithm. 1. Introduction The expression ln (1 + x) is an important expression in mathematics. It shows up surprisingly at many places [2, 4, 5, 6]. Generally, we approximate ln (1 + x) through a finite sum of an infinite series [3]. But, the number of terms in a finite sum can make the algebra very complicated. Thus, we develop some simple but robust closed form approximations to ln (1 + x) through quadrature rules. The Taylor series expansion of ln (1 + x) is X1 xi (1) ln(1 + x) = (¡1)i¡1 for ¡ 1 < x · 1: i i=1 First year undergraduate students are exposed to concepts of limits and quad- rature. Through these concepts, we develop closed form approximations. The work presented in this paper is crucial for enhancing concepts such as convergence and approximation of undergraduate students. The work will encourage students to formulate other relations for representing ln (1 + x) by using other quadrature rules, for example, Lobatto [9, 11]. Based on our work, teacher can ask students to formulate even better approximations to the mathematical expression ln (1 + x). Figure 1 presents a graph of the function 1=x. The area under the graph and between the vertical lines x = n and x = n + 1 is given as Z n+1 1 dx: n x 1 The exact value of this integral is ln (1 + n ). For forming various closed form approximations, we will approximate this integral through different quadrature rules. 8 S. K. Khattri Fig. 1 2. Approximation through trapezoidal quadrature rule Trapezoidal quadrature rule [14] is given as Z n+1 1 h dx ¼ [f(x1) + f(x2)] : n x 2 Here, h = 1, x1 = n and x2 = n + 1. Thus, · ¸ 1 1 1 [ln x]n+1 ¼ + ; n 2 n n + 1 µ ¶ · 1 ¸ · 0:5 ¸ 1 1 n + 2 1 1 + n ln 1 + ¼ = 1 : n n n + 1 n 1 + n 1 Now replacing n by x in the above equation we get the expression of ln (1 + x) through the trapezoidal rule · ¸ 1 + 0:5 x (2) ln (1 + x) ¼ x : 1 + x Let us call this expression, the Trapezoidal Euler’s Log (TELOG). New close form approximations of ln (1 + x) 9 3. Approximation through Simpson’s quadrature rule 1 The Simpson’s 3 -quadrature rule [14] for approximating integrals is given as Z n+1 1 h dx ¼ [f(x0) + 4f(x1) + f(x2)] : n x 3 1 2n+1 Here, h = 2 , x0 = n, x1 = 2 and x2 = n + 1, Z · ¸ n+1 1 1 1 8 1 dx ¼ + + ; n x 6 n 2n + 1 n + 1 µ ¶ · 8 1 ¸ 1 1 1 n n (3) ln 1 + ¼ + 1 + 1 : n 6 n 2 + n 1 + n 1 Now replacing n by x in the above equation we get the expression for ln (1 + x) through the Simpson’s quadrature rule as · ¸ 1 8x x (4) ln (1 + x) ¼ x + + : 6 2 + x 1 + x 1 1 Let us call this expression the 3 -Simpson Euler’s Log ( 3 -SELOG). 3 4. Aproximation through Simpson’s 8 -quadrature rule 3 Approximation of the integral through Simpson’s 8 -quadrature rule [14] is Z n+1 1 3h dx ¼ [f(x0) + 3f(x1) + 3f(x2) + f(x3)] : n x 8 1 3n+1 3n+2 Here, h = 3 , x0 = n, x1 = 3 , x2 = 3 and x3 = n + 1 Z · ¸ n+1 1 1 36n3 + 54n2 + 20n + 1 dx ¼ 3 2 2 ; n x n (36n + 54n + 20n + 1) + 18n + 24n + 7 µ ¶ · 9 9 1 ¸ 1 1 1 n n n (5) ln 1 + ¼ + 1 + 2 + 1 : n 8 n 3 + n 3 + n 1 + n 1 Now replacing n by x in the above equation we get the expression of ln (1 + x) through the Simpson’s quadrature rule · ¸ 1 9x 9x x (6) ln (1 + x) ¼ x + + + : 8 3 + x 3 + 2x 1 + x 3 3 Let us call this expression the 8 -Simpson Euler’s Log ( 8 -SELOG). 5. Approximation through Boole’s quadrature rule The Boole’s quadrature rule [13] is Z n+1 1 2h dx ¼ [7f(x0) + 32f(x1) + 12f(x2) + 32f(x3) + 7f(x4)] : n x 45 10 S. K. Khattri 1 4n+1 4n+2 4n+3 Here, h = 4 , x0 = n, x1 = 4 , x2 = 4 , x3 = 4 and x4 = n + 1, µ ¶ · ¸ 1 2 7 32 £ 4 12 £ 4 32 £ 4 7 ln 1 + ¼ + + + + : n 4 £ 45 n 4n + 1 4n + 2 4n + 3 n + 1 1 Now replacing n by x in the above equation we get the expression of ln (1 + x) through the Boole’s rule · ¸ 1 128x 48x 128x 7x (7) ln (1 + x) ¼ 7x + + + + : 90 4 + x 4 + 2x 4 + 3x 1 + x Let us call this expression the Boole Euler’s Log (BELOG). 6. Approximation through Gauss-Legendre 2 point quadrature The two point Gauss-Legendre Quadrature [12] is Z n+1 1 dx ¼ k [w1 f(x1) + w2 f(x2)] : n x Here, k = n+1¡n = 1 , x = 2n+1 + p1 and x = 2n+1 ¡ p1 . Weights are w = 1 2 2 1 2 2 3 3 2 2 3 1 and w2 = 1. Z " p p # n+1 1 1 2 3 2 3 dx ¼ p + p ; n x 2 (2n + 1) 3 + 1 (2n + 1) 3 ¡ 1 µ ¶ 6 3 1 6n + 3 n + n2 ln 1 + ¼ 2 = 6 1 n 6n + 6n + 1 6 + n + n2 1 Now replacing n by x in the above equation we get the expression for ln (1 + x) through the two point Gauss-Legendre quadrature rule as 6x + 3x2 (8) ln (1 + x) ¼ 6 + 6x + x2 Let us call this expression the two point Gauss-Legendre Log (2P-GLLOG). 7. Approximation through Gauss-Legendre 3 point quadrature Three point Gauss-Legendre quadrature rule [12] is given as Z n+1 1 dx ¼ k [w1 f(x1) + w2 f(x2) + w3 f(x3)] : n x The weights wi and points xi are given as 8 2n + 1 w1 = ; x1 = ; 9 2 p p 5 (2n + 1) 5 + 3 w2 = ; x2 = p ; 9 2 5 p p 5 (2n + 1) 5 ¡ 3 w3 = ; x3 = p : 9 2 5 New close form approximations of ln (1 + x) 11 Thus, µ ¶ " # 1 1 60n2 + 60n + 11 ln 1 + ¼ 6 ¡ p p p ¢ ¡ p p p ¢ n 9 (2n + 1) 2 5n + 5 + 3 2 5n + 5 ¡ 3 · 2 ¸ · 60 60 11 ¸ 60 n + 60 n + 11 n + n2 + n3 = 3 2 = 90 36 3 : 60 n + 90 n + 36 n + 3 60 + n + n2 + n3 1 Now replacing n by x in the above equation we get the expression ln (1 + x) through the three point Gauss-Legendre quadrature rule as · ¸ 60x + 60x2 + 11x3 (9) ln (1 + x) ¼ 60 + 90x + 36x2 + 3x3 Let us call this definition, the three point Gauss-Legendre Log (3P-GLLOG). 8. Approximation through Gauss-Legendre 4 point quadrature The four point Gauss-Legendre quadrature rule [12] is given as Z n+1 1 dx ¼ k [w1f(x1) + w2f(x2) + w3f(x3) + w4f(x4)] : n x 1 Here, k = 2 . Weights wi and points xi are given as r p q p (2n + 1) 7 + 3 ¡ 2 6 18 + 30 5 w1 = ; x1 = p ; 36 2 7 r p q p (2n + 1) 7 ¡ 3 ¡ 2 6 18 + 30 5 w2 = ; x2 = p ; 36 2 7 r p q p (2n + 1) 7 + 3 + 2 6 18 ¡ 30 5 w3 = ; x3 = p ; 36 2 7 r p q p (2n + 1) 7 ¡ 3 ¡ 2 6 18 ¡ 30 5 w4 = ; x4 = p : 36 2 7 Thus, µ ¶ 1 420n3 + 630n2 + 260n + 25 ln 1 + ¼ n 420n4 + 840n3 + 540n2 + 120n + 6 420 630 260 25 n + n2 + n3 + n4 = 420 840 540 120 6 : 1 + n + n2 + n3 + n4 1 Now replace n by x in the above equation. The expression for ln (1 + x) through the four point Gauss-Legendre quadrature rule is 420x + 630x2 + 260x3 + 25x4 (10) ln (1 + x) ¼ 420 + 840x + 540x2 + 120x3 + 6x4 Let us call this expression the four point Gauss-Legendre Log (4P-GLLOG). 12 S. K. Khattri 9. Approximation through Gauss-Legendre 5 point quadrature The five point Gauss-Legendre quadrature rule [12] is given as Z n+1 1 dx ¼ k [w1 f(x1) + w2 f(x2) + w3 f(x3) + w4 f(x4) + w5 f(x5)] : n x 1 Here, k = 2 . Weights wi and points xi are given as 128 1 w1 = ; x1 = n + ; 225 2 q 161 13 p 1 1 p w2 = + 70; x2 = n + + 245 ¡ 14 70; 450 900 2 42 q 161 13 p 1 1 p w3 = + 70; x3 = n + ¡ 245 ¡ 14 70; 450 900 2 42 q 161 13 p 1 1 p w4 = + 70; x4 = n + + 245 + 14 70; 450 900 2 42 q 161 13 p 1 1 p w = ¡ 70; x = n + ¡ 245 + 14 70: 5 450 900 5 2 42 µ ¶ 1 7560n4 + 15120n3 + 9870n2 + 2310n + 137 ln 1 + ¼ n 7560n5 + 18900n4 + 16800n3 + 6300n2 + 900n + 30 7560 15120 9870 2310 137 n + n2 + n3 + n4 + n5 = 7560 18900 16800 6300 900 30 : 1 + n + n2 + n3 + n4 + n5 1 Now replacing n by x in the above equation the expression for ln (1 + x) through the five point Gauss-Legendre quadrature rule is obtained 7560x + 15120x2 + 9870x3 + 2310x4 + 137x5 (11) ln (1 + x) ¼ 7560 + 18900x + 16800x2 + 6300x3 + 900x4 + 30x5 Let us call this expression the five point Gauss-Legendre Log (5P-GLLOG).