Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Questions to Guide Your Review - Page 635: 6

Answer

See below.

Work Step by Step

Given a sequence of numbers $\{a_{n}\}$, an expression of the form $\displaystyle \sum_{n=1}^{\infty}a_{n}=a_{1}+a_{2}+a_{3}+\cdots+a_{n}+\cdots$ is an infinite series. The number $a_{n}$ is the $n^{th}$ term of the series. The n-th partial sum of the series is an element of the sequence $s_{n}=\displaystyle \sum_{k=1}^{n}a_{k}=a_{1}+a_{2}+a_{3}++a_{n}$ If the sequence of partial sums $\{s_{n}\} $ has a limit $L$, then we say that the series converges to the sum $L$ and we write $\displaystyle \sum_{n=1}^{\infty}a_{n}=L$. If the sequence of partial sums of the series does not converge, we say that the series diverges. Example of a convergent series $\displaystyle \sum_{n=1}^{\infty}\frac{1}{2^{n}}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\quad=1$ Example of a divergent series: $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$
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