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Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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How do you do the Alternating Series Test on this series and what is the result? Please show work so I can follow along.

00 (-1)" n n + 1 n=2
Transcribed Image Text:00 (-1)" n n + 1 n=2
Expert Solution
Step 1

Alternating series:

A series of the form

{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots \!}

where either all an are positive or all an are negative, is called an alternating series.

The alternating series test then says: if {\displaystyle |a_{n}|} decreases monotonically. and {\displaystyle \lim _{n\to \infty }a_{n}=0} then the alternating series converges.

Moreover, let L denote the sum of the series, then the partial sum

{\displaystyle S_{k}=\sum _{n=0}^{k}(-1)^{n}a_{n}\!}

approximates L with error bounded by the next omitted term:

{\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!}
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