# How to determine is sum(-1)^n n/(n+1) absolutely convergent, conditionally convergent or neither?

Mar 28, 2018

neither

#### Explanation:

${\lim}_{n \rightarrow \infty} {\left(- 1\right)}^{n} \cdot \frac{n}{n + 1} \ne 0$

diverges by nth term test

Mar 28, 2018

Diverges

#### Explanation:

This is an alternating series. Let us first attempt to try the alternating series test, which tells us if we have a series in the form

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n} {b}_{n}$ and ${\lim}_{n \to \infty} {b}_{n} = 0 , {b}_{n} \le {b}_{n + 1} ,$ the series converges.

Here,

${b}_{n} = \frac{n}{n = 1} , {\lim}_{n \to \infty} \frac{n}{n + 1} = 1 \ne 0$

So, the alternating series test is inconclusive.

Instead, we'll use the Divergence Test, and take

${\lim}_{n \to \infty} {\left(- 1\right)}^{n} \cdot \frac{n}{n + 1}$

This limit does not truly exist due to the ${\left(- 1\right)}^{n}$, but since $\frac{n}{n + 1}$ is always getting closer to one, we convince ourselves the terms are alternating signs and approaching one, and so the series diverges.