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A better remainder Suppose an alternating series .
Chapter 10, Problem 54AE(choose chapter or problem)
A better remainder Suppose an alternating series \(\sum_{k=1}^{\infty}(-1)^k\ a_k\) converges to S and the sum of the first n terms of the series is \(S_{n}\). Suppose also that the difference between the magnitudes of consecutive terms decreases with k. Then it can be shown that for \(n \geq 1\)
\(\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|\)
a. Interpret this inequality and explain why it gives a better approximation to S than simply using \(S_{n}\) to approximate S.
b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than \(10^{-6}\) using both \(S_{n}\) and the method explained in part (a).
(i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\)
Questions & Answers
QUESTION:
A better remainder Suppose an alternating series \(\sum_{k=1}^{\infty}(-1)^k\ a_k\) converges to S and the sum of the first n terms of the series is \(S_{n}\). Suppose also that the difference between the magnitudes of consecutive terms decreases with k. Then it can be shown that for \(n \geq 1\)
\(\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|\)
a. Interpret this inequality and explain why it gives a better approximation to S than simply using \(S_{n}\) to approximate S.
b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than \(10^{-6}\) using both \(S_{n}\) and the method explained in part (a).
(i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\)
ANSWER:Problem 54AEA better remainder Suppose an alternating series . converges to S and the sum of the first n terms of the series is Sn. Suppose also that the difference between the magnitudes of consecutive terms decreases with k. Then it can be shown that for n 1 .a. Interpret this inequality and explain why it gives a better approximation to S than simply using Sn to approximate S.b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than 106 using both Sn and the method explained in part (a).(i) (ii) (iii) Answer ; Step-1 ; Given is ; A better remainder suppose an alternating series . Converges to S and the sum of the first n terms of the series . Suppose also that the difference between the magnitudes of consecutive terms decreases with k . Then it can be shown