CUBO A Mathematical Journal Vol.21, No¯ 02, (51–64). August 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000200051 The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series Vito Lampret University of Ljubljana, 386 Slovenia vito.lampret@guest.arnes.si ABSTRACT For the perimeter P(a, b) of an ellipse with the semi-axes a ≥ b ≥ 0 a sequence Qn(a, b) is constructed such that the relative error of the approximation P(a, b) ≈ Qn(a, b) satisfies the following inequalities 0 ≤ − P(a, b) − Qn(a, b) P(a, b) ≤ (1 − q2)n+1 (2n + 1)2 ≤ 1 (2n + 1)2 e−q 2(n+1), true for n ∈ N and q = b a ∈ [0, 1]. RESUMEN Para el peŕımetro P(a, b) de una elipse con semiejes a ≥ b ≥ 0, se construye una sucesión Qn(a, b) tal que el error relativo de la aproximación P(a, b) ≈ Qn(a, b) satisface las siguientes desigualdades 0 ≤ − P(a, b) − Qn(a, b) P(a, b) ≤ (1 − q2)n+1 (2n + 1)2 ≤ 1 (2n + 1)2 e−q 2(n+1), válidas para n ∈ N y q = b a ∈ [0, 1]. Keywords and Phrases: approximation, elementary, ellipse, estimate, Maclaurin series, mathe- matical validity, perimeter, simple. 2010 AMS Mathematics Subject Classification: 40A25, 65B10. http://dx.doi.org/10.4067/S0719-06462019000200051 52 Vito Lampret CUBO 21, 2 (2019) 1 Introduction Injective parametric equations of the border of an ellipse having semi-axes of lengths a and b ≤ a are given as x = x(t) = a cos(t), y = y(t) = b sin(t), where t ∈ [0, 2π). Its perimeter P(a, b) is determined as P(a, b) = ∫ 2π 0 √ ẋ2(t) + ẏ2(t) dt = 4 ∫ π 2 0 √ a2 sin2(t) + b2 cos2(t) dt = 4a ∫ π 2 0 √ 1 − ǫ2 cos2(t) dt = ︷ ︸︸ ︷ t = π/2 − τ 4a ∫ 0 π 2 √ 1 − ǫ2 sin2(τ)(− dτ). Thus, the perimeter P(a, b) of an ellipse having semi-axes of lengths a and b ≤ a, is given as P(a, b) = 4a E(ǫ), (1.1) where E(ǫ) := ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ (1.2) is complete elliptic integral of the second kind and ǫ := √ 1 − ( b a )2 = √ a2−b2 a2 ∈ [0, 1), (1.3) is the eccentricity of an ellipse. For b ≈ 0 it is intuitively evident that P(a, b) > 2 × 2a = 4a. Moreover, since the functions ǫ 7→ 1 − ǫ2 sin2(τ) are decreasing on the interval [0, 1] for any τ ∈ [0, π/2], the function E(ǫ) is decreasing too. Therefore, we have 1 = ∫ π 2 0 cos(τ) dτ = E(1) ≤ E(ǫ) ≤ E(0) = π 2 , for 0 ≤ ǫ ≤ 1. Consequently, due to (1.1), inf 0 n; consequently w0 = 1 54 Vito Lampret CUBO 21, 2 (2019) we obtain ( 1 2 i ) = ∏i−1 j=0( 1 2 − j) i! = (−1)i−1 1 2i · ∏i−1 j=1(2j − 1) ∏i j=1 j = (−1)i−1 1 2i − 1 i ∏ j=1 2j − 1 2j = (−1)i−1 wi 2i − 1 . (2.3) Thus, thanks to (2.1), replacing x by −x, we get (1 − x) 1 2 = 1 − n∑ i=1 wi 2i − 1 xi + rn(x), (2.4) with the remainder rn(x) = −x n+1 wn+1 2n + 1 (n + 1) ∫ 1 0 ( 1 − t 1 − tx )n dt (1 − tx) 1 2 , estimated, for x ∈ (0, 1), as 0 < −rn(x) = xn+1 (1 − x) 1 2 · wn+1 2n + 1 (n + 1) ∫ 1 0 ( 1 − t 1 − tx )n dt < wn+1 (1 − x) 3 2 (2n + 1) xn+1 . (2.5) Indeed, using the substitution τ = 1−t 1−tx , i.e. t = 1−τ 1−τx we have (considering x ∈ (0, 1)) ∫ 1 0 ( 1 − t 1 − tx )n dt = ∫ 0 1 τ n ( − 1 − x (1 − τx)2 ) dτ = ∫ 1 0 τ n · 1 − x (1 − τx)2 dτ < ∫ 1 0 τ n · 1 − x (1 − x)2 dτ = 1 (1 − x)(n + 1) . 2.2 Wallis ratios estimates The integrals In := ∫ π 2 0 sinn(x) dx (n ≥ 0), (2.6) satisfy the recurrence relation In = n − 1 n In−2, for n ≥ 2, where, obviously, we have I0 = π 2 and I1 = 1. Consequently, by induction we find I2i = i ∏ j=1 2j − 1 2j · π 2 = wi · π 2 (2.7) CUBO 21, 2 (2019) The perimeter of a flattened ellipse can be estimated accurately . . . 55 and I2i+1 = i∏ j=1 2j 2j + 1 = 1 (2i + 1)wi . (2.8) Obviously, we estimate 0 < sin2i+2(x) < sin2i+1(x) < sin2i(x) < 1, for x ∈ ( 0, π 2 ) and i ∈ N. Integrating, we obtain 0 < I2i+2 < I2i+1 < I2i < 1, for all i ∈ N. Hence, thanks to (2.7)–(2.8), we get 2i + 1 2i + 2 wi · π 2 = wi+1 · π 2 < 1 (2i + 1)wi < wi · π 2 . Consequently, 2 π · 1 2i + 1 < w2i < 2 π · 1 2i − 1 (i ∈ N). (2.9) We remark that there exists a huge literature on useful, more accurate estimates for wn, e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 31]. However, for our needs, there suffice rather rough estimates (2.9). 2.3 Some logarithmic formula expansion For p ≥ 1 and −1 < t < 1 we have 2 p−1 ∑ j=0 t2j = 2(p−1) ∑ k=0 ( tk + (−t)k ) = 2(p−1) ∑ k=0 tk + 2(p−1) ∑ k=0 (−t)k = 1 − t2p−1 1 − t + 1 − (−t)2p−1 1 + t . Consequently, integrating from 0 to x ∈ (−1, 1), the first and the last part of these equalities, we obtain 2 p−1 ∑ j=0 x2j+1 2j + 1 = ∫ x 0 1 1 − t dt − ∫ x 0 t2p−1 1 − t dt + ∫ x 0 1 1 + t dt + ∫ x 0 t2p−1 1 + t dt = − ln(1 − x) + ln(1 + x) − ∫ x 0 ( 1 1 − t − 1 1 + t ) t2p−1 dt ︸ ︷︷ ︸ =R∗ p (x) . Thus, ln ( 1 + x 1 − x ) = 2 p ∑ i=1 x2i−1 2i − 1 + R∗p(x), (2.10) 56 Vito Lampret CUBO 21, 2 (2019) with the remainder R∗p(x), R∗p(x) := ∫ x 0 2t2p 1 − t2 dt ≥ ∫ x 0 2t2p dt. (0 < x < 1), estimated as 2x2p+1 2p + 1 < R∗p(x) < 2x2p+1 (1 − x2)(2p + 1) (p ∈ N, 0 < x < 1) (2.11) From (2.10)–(2.11) we end up with the expansion ln ( 1 + x 1 − x ) = 2 ∞ ∑ i=1 x2i−1 2i − 1 , (2.12) true for x ∈ (0, 1) and, consequently, also for x ∈ (−1, 0]. 3 The Maclaurin series 3.1 Derivation Referring to (2.4)–(2.5) and (2.6)–(2.7), we have, for any n ∈ N, ∫ π 2 0 √ 1 − ǫ2 sin2(τ) ︸ ︷︷ ︸ dτ = π 2 − n ∑ i=1 wi ǫ 2i 2i − 1 ∫ π 2 0 sin2i(τ) dτ + r∗n(ǫ) = π 2 − n ∑ i=1 wi ǫ 2i 2i − 1 ( wi π 2 ) + r∗n(ǫ). Hence ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π 2 ( 1 − n ∑ i=1 w2i 2i − 1 ǫ2i ) + r∗n(ǫ), (3.1) where wi is the i-th Wallis’ ratio and the error term r ∗ n(ǫ) := ∫ π/2 0 rn ( ǫ2 sin2(τ) ) dτ is estimated, due to (2.5) and considering (2.6)–(2.7), as 0 ≤ −r∗n(ǫ) ≤ ǫ2n+2 1 − ǫ2 · wn+1 2n + 1 ∫ π 2 0 sin2n+2(τ) dτ = ǫ2n+2 wn+1 (1 − ǫ2)(2n + 1) · wn+1 π 2 . Thus, according to (2.9), 0 ≤ −rn(ǫ) ≤ π 2 · 1 1 − ǫ2 · w2n+1 2n + 1 ǫ2n+2 ≤ 1 1 − ǫ2 · ǫ2n+2 (2n + 1)2 . (3.2) This estimate is not usable for ǫ ≈ 1, i.e. for b ≈ 0 (for a very flattened ellipse). CUBO 21, 2 (2019) The perimeter of a flattened ellipse can be estimated accurately . . . 57 As w2n ≤ 1, we have lim n→∞ rn(ǫ) = 0 for any ǫ < 1, which is always true for ordinary ellipse, due to the equivalence ǫ = 1 ⇔ b = 0. Hence, there holds the so-called Maclaurin series expansion5 ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π 2 ( 1 − ∞ ∑ i=1 w2i 2i − 1 ǫ2i ) , (3.3) valid for 0 ≤ ǫ < 1. In addition, the series on the right is convergent also for ǫ = 1 due to (2.9). Indeed, we have w2i 2i−1 < 1 i2 , which implies the convergence of the series ∑∞ i=1 w2i 2i−1 . Remark 3.1. About fifty years after Maclaurin’s book [24], including the series (3.3), Ivory pub- lished article [13], where he presented the expansion ∫ π 2 0 √ 1 − ǫ2 sin2(τ) dτ = π(a + b) 4a ( 1 + ∞∑ i=1 w2i (2i − 1)2 λ2i ) ( λ = a − b a + b ) , where the series on the right converges slightly faster than the series in (3.3). Applying (2.9) for the partial sums µn(ǫ) := n ∑ i=1 w2i 2i − 1 ǫ2i (n ∈ N ∪ {∞}), (3.4) we shall estimate the series µ∞(ǫ) figuring in (3.3). 3.2 Approximating µ ∞ (ǫ) Using (2.9) we estimate, 2 π(2i − 1)(2i + 1) < w2i 2i − 1 < 2 π(2i − 1)2 (i ∈ N) . (3.5) Therefore µ∞(ǫ) ≈ ∞ ∑ i=1 2ǫ2i π(2i − 1)(2i + 1) (0 ≤ ǫ < 1). This idea produces the next theorem. Theorem 3.2. We have µ∞(ǫ) = Mn(ǫ) + δn(ǫ), (3.6) where Mn(ǫ) = A(ǫ) + Bn(ǫ), (3.7) A(ǫ) := 1 2π [( ǫ − 1 ǫ ) ln (1 + ǫ 1 − ǫ ) + 2 ] ∈ ( 0, 1 π ) , (3.8) 5The coefficients of the original Maclaurin series [24] have a visually more complicated form. 58 Vito Lampret CUBO 21, 2 (2019) Bn(ǫ) := n ∑ i=1 ( w2i − 2 π(2i + 1) ) ǫ2i 2i − 1 , (3.9) and 0 < δn(ǫ) < δ ∗ n(ǫ) := 2ǫ2n+2 π(2n + 1)2 , (3.10) valid for any integer n ≥ 1 and every 0 < ǫ < 1. The basic function A(ǫ) is strictly increasing having the range ( 0, 1 π ) , where lim ǫ↓0 A(ǫ) = 0 and lim ǫ↑1 A(ǫ) = 1 π . Both sequences, n 7→ Bn(ǫ) and n 7→ δn(ǫ), are strictly increasing, for every ǫ ∈ (0, 1). The sequence n 7→ Mn(ǫ) converges strictly increasingly to µ∞(ǫ), for any ǫ ∈ (0, 1). Addi- tionally, for every n ∈ N, the functions ǫ 7→ Mn(ǫ) and ǫ 7→ δn(ǫ) are strictly increasing on the interval (0, 1). Figure 1 shows, on the left, the graph6 of the basic function A(ǫ), and, on the right, the graphs of the functions M∗1 (ǫ) and µ∞(ǫ). As an example, we present B ∗ 4(ǫ) and δ ∗ 4(ǫ) as follows: B∗4(ǫ) = ( 1 4 − 2 3π ) ǫ2 + 1 3 ( 9 64 − 2 5π ) ǫ4 + 1 5 ( 25 256 − 2 7π ) ǫ6 + 1 7 ( 1225 16384 − 2 9π ) ǫ8 ≈ 0.037 793 409 ǫ2 + 0.004 433 682 ǫ4 + 0.001 342 114 ǫ6 + 0.000 576 077 ǫ8, δ∗4(ǫ) ≤ 2ǫ10 81π ≤ 0.00786 ǫ10 ( ǫ = √ 1 − ( b a )2 ) . 0.2 0.4 0.6 0.8 1.0 0.05 0.10 0.15 0.20 0.25 0.30 AHΕL 0.2 0.4 0.6 0.8 1.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 AHΕL M1 HΕL» Μ¥HΕL Figure 1: The graph of the basic function A(ǫ) (left) and the graphs of the functions M1(ǫ), µ∞(ǫ) and A(ǫ) (right). Proof of Theorem 3.2. We have, for 0 < ǫ < 1, ∞ ∑ i=1 w2i ǫ2i 2i − 1 = ∞ ∑ i=1 2 ǫ2i π(2i − 1)(2i + 1) (3.11) + n ∑ i=1 ( w2i 2i − 1 − 2 π(2i − 1)(2i + 1) ) ǫ2i + δn(ǫ), 6All the graphics and calculations in this paper are made using the Mathematica [30] computer system. CUBO 21, 2 (2019) The perimeter of a flattened ellipse can be estimated accurately . . . 59 where δn(ǫ) = ∞ ∑ i=n+1 ( w2i − 2 π(2i + 1) ) ǫ2i 2i − 1 . (3.12) Moreover, using (2.12), we have ∞ ∑ i=1 2 π(2i − 1)(2i + 1) ǫ2i = 1 π ∞ ∑ i=1 ( 1 2i − 1 − 1 2i + 1 ) ǫ2i = 1 π ( ǫ 2 · 2 ∞ ∑ i=1 ǫ2i−1 2i − 1 − 1 2ǫ · 2 ∞ ∑ i=1 ǫ2i+1 2i + 1 ) = 1 π [ ǫ 2 ln (1 + ǫ 1 − ǫ ) − 1 2ǫ ( ln (1 + ǫ 1 − ǫ ) − 2ǫ )] = 1 2π [( ǫ − 1 ǫ ) ln (1 + ǫ 1 − ǫ ) + 2 ] = A(ǫ). Concerning A(ǫ) = 1 2π ( f(ǫ) + 2 ) , the function f(ǫ) := ( ǫ − 1 ǫ ) ln ( 1+ǫ 1−ǫ ) (0 < ǫ < 1) has the derivative f′(ǫ) = g(ǫ)/ǫ2, where g(ǫ) = (1 + ǫ2) ln ( 1+ǫ 1−ǫ ) − 2ǫ, having the derivative g′(ǫ) = 2ǫ 1 − ǫ2 ( 2ǫ + (1 − ǫ2) ln ( 1 + ǫ 1 − ǫ )) > 0 (0 < ǫ < 1). Thus, g is strictly increasing on [0, 1). Consequently, g(ǫ) > g(0) = 0, i.e. f′(ǫ) > 0, for 0 < ǫ < 1. Therefore, f is strictly increasing on (0, 1) too. Moreover, using (2.10)–(2.11) with p = 1, we have f(ǫ) = ǫ 2 −1 ǫ · 2 ( ǫ + ϑ · 2ǫ 3 3(1−ǫ2) ) = 2(ǫ2 − 1) ( 1 + ϑ · 2 1−ǫ2 · ǫ 2 3 ) , for some ϑ = ϑ(ǫ) ∈ (0, 1). Hence, lim ǫ↓0 f(ǫ) = −2, i.e. lim ǫ↓0 A(ǫ) = lim ǫ↑1 1 2π ( f(ǫ) + 2 ) = 0. In addition, lim ǫ↑1 f(ǫ) = lim ǫ↑1 [ ǫ2−1 ǫ · 2 ln(1 + ǫ) ] − 1 1 · lim h↓0 ( − h ln(h) ) = 0, where h = 1 − ǫ2. Thus, lim ǫ↑1 A(ǫ) = lim ǫ↑1 1 2π ( f(ǫ) + 2 ) = 1 π . According to (3.5), all summands in Bn(ǫ) and δn(ǫ) (see (3.12)) are positive, i.e. the sequences n 7→ Bn(ǫ) and n 7→ δn(ǫ) are strictly increasing; consequently the sequence n 7→ Mn(ǫ) is also strictly increasing, for every ǫ ∈ (0, 1). Since all coefficients of the power series Bn(ǫ) and δn(ǫ) (see (3.9) and (3.12)) are positive, due to (3.5), the functions ǫ 7→ Mn(ǫ) and ǫ 7→ δn(ǫ) are strictly increasing on the interval (0, 1), for any n ∈ N. According to (3.12) and (3.5), we estimate, for ǫ ∈ (0, 1], 0 < δn(ǫ) < ∞ ∑ i=n+1 ( 2 π(2i − 1) − 2 π(2i + 1) ) ǫ2n+2 2n + 1 = 2ǫ2n+2 π(2n + 1)2 , 60 Vito Lampret CUBO 21, 2 (2019) using the telescoping method of summation. Example 3.3. Theorem 3.2 is quite useful for an estimate of µ∞(ǫ), consequently for an esti- mate of the perimeter of an ellipse. For example, for a very flattened ellipse with q = 0.01 we have 0.99994 < ǫ(q) < 0.99995 where 0.36315 < M20(0.99995) < 0.36316 . . . and δ ∗ 20(0.99995) < 0.00038. Therefore, 0.36315 < µ∞(0.99995) < 0.36316+0.00038 = 0.36354. Thus, to three decimal places, we have µ∞(0.99995) = 0.363 . . .. Consequently, the perimeter P(a, b) of the corresponding ellipse is given as P(a, b) = 4a · π 2 ( 1 − µ∞(0.99995) ) ≈ 4a · π 2 ( 1 − 0.363 ) ≈ 4.002 a (compare with relations (1.4)). Remark 3.4. Referring to Abel’s theorem on the boundary behavior of a power series, if we continuously extend the domain of the function A(ǫ) to the closed interval [0, 1] by using limits, A(0) := 0 and A(1) := 1 π , then (3.6), (3.7), (3.9) and (3.10) are all valid also for ǫ = 0 and ǫ = 1. Remark 3.5. Alternatively, we can estimate the remainder r∗∗n (ǫ) := µ∞(ǫ) − Mn(ǫ) as follows: r∗∗n (ǫ) ≤ ∞ ∑ i=n+1 w2i ǫ 2i 2i − 1 ≤ w2n+1ǫ 2n+2 2n + 1 ∞ ∑ j=0 ǫ2j = w2n+1ǫ 2n+2 (2n + 1)(1 − ǫ2) ≤ 1 1 − ǫ2 · 2ǫ2n+2 π(2n + 1)2 . This simple method works quite well for ǫ, which “differs enough from 1”, but it is useless for ǫ, which is close to 1. 4 The main result Theorem 4.1. For every n ∈ N, the perimeter P(a, b) of an ellipse having semi-major and semi- minor axes, a and b, the aspect ratio q = q(a, b) := b/a, and the eccentricity ǫ = ǫ(a, b) := √ 1 − q2, the n-th approximation Qn(a, b) ≈ P(a, b), Qn(a, b) := 2πa ( 1 − Mn ( ǫ ) ) = 2πa ( 1 − A(ǫ) − Bn ( ǫ ) ) , (4.1) has the relative error, P(a, b) − Qn(a, b) P(a, b) =: ρn(q) ( q = q(a, b) = ( b a )2 ) , estimated as − 1 (2n + 1)2 e−q 2(n+1) ≤ − ( 1 − q2 )n+1 (2n + 1)2 =: ρ∗n(q) ≤ ρn(q) ≤ 0 . Here, A(ǫ) and Bn ( ǫ ) are defined in Theorem 3.2 and we have Bn+1 ( ǫ ) = Bn ( ǫ ) + ( w2n+1 − 2 π(2n+3) ) ǫ2n+2 2n+1 , for n ∈ N and 0 ≤ ǫ ≤ 1. CUBO 21, 2 (2019) The perimeter of a flattened ellipse can be estimated accurately . . . 61 Proof. Thanks to (1.1), (1.2), (1.4) and (3.3), we estimate − P(a, b) − Qn(a, b) P(a, b) = − 2πa ( 1 − Mn(ǫ) − δn(ǫ) ) − 2πa ( 1 − Mn(ǫ) ) P(a, b) (1.4) < 2πa δn(ǫ) 4a ≤ π δn(ǫ) 2 < π 2 · 2ǫ2n+2 π(2n + 1)2 = ǫ2n+2 (2n + 1)2 , where, considering the convexity of the exponential function or, referring to [16, (6a)] with ε = q2 and h = −q2 , we have ǫ2n+2 = (1 − q2)n+1 ≤ e−q 2(n+1) (0 ≤ q < 1). Figures 2–3 show, for several values of n, the graphs of actual relative errors −ρn(q) = [ µ∞ ( ǫ(q) ) − Mn ( ǫ(q) )] / [ 1 − µ∞ ( ǫ(q) )] (left) together with their upper bounds −ρ∗n(q) (right). 0.2 0.4 0.6 0.8 1.0 0.002 0.004 0.006 0.008 0.010 -Ρ1HqL 0.2 0.4 0.6 0.8 1.0 0.02 0.04 0.06 0.08 0.10 -Ρ1 * HqL Figure 2: The graphs of the functions q 7→ −ρ1(q) and q 7→ −ρ ∗ 1(q). 0.2 0.4 0.6 0.8 1.0 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 -Ρ9HqL 0.2 0.4 0.6 0.8 1.0 0.0005 0.0010 0.0015 0.0020 0.0025 -Ρ9 * HqL Figure 3: The graphs of the functions q 7→ −ρ9(q) and q 7→ −ρ ∗ 9(q). Table 1 additionally confirms the usefulness of the derived formula. Conclusion. The article demonstrates that with the help of 277 years old Maclaurin series the perimeter of an ellipse can be accurately estimated, even if an ellipse flattens into a line segment. This is done only by elementary means, not using complex analysis or elliptical integral theory, neither arithmetic-geometric means nor hypergeometric functions. 62 Vito Lampret CUBO 21, 2 (2019) q 0.00001 0.1 0.2 0.3 0, 4 0, 5 −ρ20(q) < 8·10 −5 < 6·10−5 < 2·10−5 < 5·10−6 < 6·10−7 < 4·10−8 −ρ∗20(q) < 6·10 −4 < 5·10−4 < 3·10−4 < 9·10−5 < 2·10−5 < 2·10−6 Table 1: The actual error ρ20(q) and the a priori estimated error ρ ∗ 20(q). References [1] S. Adlaj, An eloquent formula for the perimeter of an ellipse, Notices of the AMS 59 (2012), no. 8, 1094–1099. [2] G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608. [3] B. W. 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Zhang, T. Q. Xu and L. B. Situ Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007) 1, art. 17, 9 p. Introduction Background The binomial approximation Wallis ratios estimates Some logarithmic formula expansion The Maclaurin series Derivation Approximating () The main result