"Use the alternating series test to study the..." solved and explained

Marvin Mccormick

Answered question

2021-02-02

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

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

Answer & Explanation

escumantsu

Skilled2021-02-03Added 98 answers

Alternating series:
A series of the form
$\sum_{n = 1}^{\infty} (- 1)^{n + 1} n a_{n} = a_{0} - a_{1} + a_{2} - a_{3} + . . .$
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:
$| S_{k} - L | \leq | S_{k} - S_{k + 1} | = a_{k + 1}$
i) $a_{n} = n e^{- n}$
$= \frac{n}{e^{n}} > 0$
ii) $n \to \infty \frac{n}{e^{n}}$
$n \to \infty \frac{1}{e^{n}}$
$= \frac{1}{\infty}$
$= 0$
iii) $a_{n + 1}$

Hence the alternating series will converge.

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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