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Consider the following family of series, one for each constant p > 0: 1 n(logn) S(p) = For each N > 2 we define the partial sum up to index N and the corresponding remainder as follows: SN (P) = N 1 n(logn)P (a) Express the following definite integral in terms of a, b, and p (assume a > 1): dx Sº x(log x) P RN(P) SSN(p). = a1 In(b)^(1-p)/(1-p)-In(a)^(1-p)/(1-P) Use p = 2.8 in all parts below. Please enter your answers in calculator ready form. (b) Express the following definite integral in terms of a (assume a > 1): Soz dx 1/(1.8ln^(1.8)(a)) x(log x) 2.8 (c) The integral test provides a two-sided inequality of the following form for RN (2.8) when N = 45: DO da < R4s (2.8) ≤2(log 2)2.8 用 Soz dx x(log x) 2.8 Find the integers a₁ and a₂ above. (Use the smallest compatible value of a₁ and the largest compatible value of a 2.) a2

Consider the following family of series, one for each constant p > 0: 1 n(logn) S(p) = For each N > 2 we define the partial sum up to index N and the corresponding remainder as follows: SN (P) = N 1 n(logn)P (a) Express the following definite integral in terms of a, b, and p (assume a > 1): dx Sº x(log x) P RN(P) SSN(p). = a1 In(b)^(1-p)/(1-p)-In(a)^(1-p)/(1-P) Use p = 2.8 in all parts below. Please enter your answers in calculator ready form. (b) Express the following definite integral in terms of a (assume a > 1): Soz dx 1/(1.8ln^(1.8)(a)) x(log x) 2.8 (c) The integral test provides a two-sided inequality of the following form for RN (2.8) when N = 45: DO da < R4s (2.8) ≤2(log 2)2.8 用 Soz dx x(log x) 2.8 Find the integers a₁ and a₂ above. (Use the smallest compatible value of a₁ and the largest compatible value of a 2.) a2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following family of series, one for each constant p > 0: S(p) = n=2 For each N > 2 we define the partial sum up to index N and the corresponding remainder as follows: 1 n=2 n(logn)p¹ (a) Express the following definite integral in terms of a, b, and p (assume a > 1): dx S. (log z)P SN (P) = 1 n(logn) ∞ a1 = = Use p = 2.8 in all parts below. Please enter your answers in calculator ready form. (b) Express the following definite integral in terms of a (assume a > 1): dx zloga) 28 RN (P) SSN (P). S(2.8) = = In(b)^(1-p)/(1-p)-In(a)^(1-p)/(1-P) = (c) The integral test provides a two-sided inequality of the following form for RN (2.8) when N = 45: dx So · ≤ R$45 (2.8) ≤ 0 x(log x) 2.8 441 Find the integers a₁ and a2 above. (Use the smallest compatible value of a₁ and the largest compatible value of a 2.) 1/(1.8ln^(1.8)(a)) a2 = dx (log z) 2.8 (d) Computation reveals S45 (2.8) = 1.966426. Use this fact, together with your earlier work, to complete this two-sided inequality involving the exact value of the series S(2.8): <S(2.8) < (e) An experimentalist might express the two-sided inequality in part (d) by writing S(2.8) = μ±e, where is the midpoint of the interval and is the distance from μ to either endpoint. Translate your findings about the given series into experimentalists' notation:
Transcribed Image Text:Consider the following family of series, one for each constant p > 0: S(p) = n=2 For each N > 2 we define the partial sum up to index N and the corresponding remainder as follows: 1 n=2 n(logn)p¹ (a) Express the following definite integral in terms of a, b, and p (assume a > 1): dx S. (log z)P SN (P) = 1 n(logn) ∞ a1 = = Use p = 2.8 in all parts below. Please enter your answers in calculator ready form. (b) Express the following definite integral in terms of a (assume a > 1): dx zloga) 28 RN (P) SSN (P). S(2.8) = = In(b)^(1-p)/(1-p)-In(a)^(1-p)/(1-P) = (c) The integral test provides a two-sided inequality of the following form for RN (2.8) when N = 45: dx So · ≤ R$45 (2.8) ≤ 0 x(log x) 2.8 441 Find the integers a₁ and a2 above. (Use the smallest compatible value of a₁ and the largest compatible value of a 2.) 1/(1.8ln^(1.8)(a)) a2 = dx (log z) 2.8 (d) Computation reveals S45 (2.8) = 1.966426. Use this fact, together with your earlier work, to complete this two-sided inequality involving the exact value of the series S(2.8): <S(2.8) < (e) An experimentalist might express the two-sided inequality in part (d) by writing S(2.8) = μ±e, where is the midpoint of the interval and is the distance from μ to either endpoint. Translate your findings about the given series into experimentalists' notation:
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