Solved: "How do you do the Alternating Series Test..."

Lipossig

Answered question

2021-02-15

How do you do the Alternating Series Test on this series and what is the result?

\sum_{n = 2}^{\infty} \frac{(- 1)^{n}}{n + 1}

Answer & Explanation

2k1enyvp

Skilled2021-02-16Added 94 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:
$| S_{k} - L | \leq | S_{k} - S_{k + 1} | = a_{k + 1}$
$a_{n} = \frac{1}{n + 1}$
$a_{n + 1} = \frac{1}{n + 2}$

Therefore $a_{n + 1} < a_{n}$

So $a_{n}$ is monotonically decrea $\sin g$

$a_{n} > 0$ for all $n = 1, 2, 3, . . .$

$n \to \infty$

$= 0$

So the series converges.

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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