Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$

Question

Prove wether or not the following series diverges or converges:

$\sum_{n=0}^\infty {(-1)^nn\over n+1}$

I am just not sure, I know if I use the absolute value test for convergence and root test it is inconclusive.

I was then thinking of using nth term test however I have the $(-1)^n$, can I just ignore that?

Any help would be greatly appreciated.

Answer 1

Hint: What can you say about $$\lim_{n\to\infty}\frac{(-1)^nn}{n+1}?$$

Look at $$\lim_{n\to\infty}\frac n{n+1}$$ first.

Answer 2

As ${(-1)^nn\over n+1}\not\to 0$ the series does not converge.

Answer 3

Fact. Let $\{a_n\}$ be a sequence. Then $\displaystyle\lim_{n\to\infty}a_n=0$ if and only if $\displaystyle\lim_{n\to\infty}\left\lvert a_n\right\rvert=0$

Now, we wish to investigate the convergence of $\sum a_n$ where $\displaystyle a_n=(-1)^n\frac{n}{n+1}$.

To do so, let's apply the Divergence Test.

Note that $$ \lim_{n\to\infty}\left\lvert a_n\right\rvert = \lim_{n\to\infty}\left\lvert (-1)^n\frac{n}{n+1}\right\rvert = \lim_{n\to\infty}\frac{n}{n+1} = 1 $$ It follows that $\displaystyle\lim_{n\to\infty}a_n\neq0$. Hence $\sum a_n$ diverges.