Use the alternating series test to study the convergence of the following series The image displays a mathematical series expression:\[\sum_{n=1}^{\infty} (-1)^{n+1} n e^{-n}\]This expression represents an infinite series that sums terms of the form $(-1)^{n+1} n e^{-n}$ starting from $n = 1$ to infinity. In this series:- $(-1)^{n+1}$ introduces an alternating sign based on the value of $n$.- $n$ is the term index, multiplying the exponential factor.- $e^{-n}$ is the exponential decay factor, where $e$ is the base of the natural logarithm.This series can be used in various applications, including mathematics and physics, to model phenomena that exhibit alternating behavior with exponentially decreasing terms.

Alternating series: A series of the form {\displaystyle \sum _{n=0}^{\infty…

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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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Use the alternating series test to study the convergence of the following series

The image displays a mathematical series expression:

\[
\sum_{n=1}^{\infty} (-1)^{n+1} n e^{-n}
\]

This expression represents an infinite series that sums terms of the form $(-1)^{n+1} n e^{-n}$ starting from $n = 1$ to infinity. In this series:

- $(-1)^{n+1}$ introduces an alternating sign based on the value of $n$.
- $n$ is the term index, multiplying the exponential factor.
- $e^{-n}$ is the exponential decay factor, where $e$ is the base of the natural logarithm.

This series can be used in various applications, including mathematics and physics, to model phenomena that exhibit alternating behavior with exponentially decreasing terms.

Expert Solution

Step 1

Alternating series:

A series of the form

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

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

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

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

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

approximates L with error bounded by the next omitted term:

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

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