Consider the series ∑n=1∞(−1)nn2 a) Show the series converges or diverges using the alternating series test. b) Approximate the sum using the 4-th partial sum(S4) of the series. c) Calculate the maximum error between partial sum(S4) and the sum of the series using the remainder term portion of the alternating series test.

Question

Consider the series ∑n=1∞(−1)nn2  a) Show the series converges or diverges using the alternating series test.  b) Approximate the sum using the 4-th partial sum(S4) of the series.  c) Calculate the maximum error between partial sum(S4) and the sum of the series using the remainder term portion of the alternating series test.

Nathanael Webber · Accepted Answer

Given series is ∑n=1∞(−1)nn2=∑n=1∞(−1)nbn bn=1n2 Since, n2&amp;lt;(n+1)2∀n∈N 1n2&amp;gt;1(n+1)2∀n∈N bn&amp;gt;bn+1∀n∈N Therefore, sequence {bn} is decreasing sequence Also, limn→∞bn=limn→∞1n2=0 Therefore, Alternating series test, given series converges. b) 4th partial sum of the series is evaluated as follows ∑n=1∞an=∑n=1∞(−1)nn2 an=(−1)nn2 S4=a1+a2+a3+a4 S4=(−1)112+(−1)222+(−1)332+(−1)442 =−1+14−19+116=−0.7986 c) By Alternating series error estimation theorem,, Maximum error between S4 and sum S of the series is |S−S4|≤|a5| |S−S4|≤|(−1)552|=0.04 Maximum error between S4 and sum S of the series is 0.04

Jeffrey Jordon · Accepted Answer

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Consider the series sum_{n=1}^inftyfrac{(-1)^n}{n^2} a) Show the series converges or diverges using the alternating series test. b) Approximate the su

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