Convergence of an infinite series problem

Question

I am having trouble with the series

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

I want to know if it converges or not, and I´ve tried with the comparision test, the ratio test, the Leibniz test...

Any ideas?

Answer 1

First let $a_n$ be a real valued sequence then we have the equivalence $$\lim_{n\rightarrow \infty}a_n=0 \Leftrightarrow \lim_{n\rightarrow \infty}|a_n|=0$$

Here $\lim_{n\rightarrow \infty}|(-1)^n\dfrac{n}{n+1}|\not = 0$ then $\lim_{n\rightarrow \infty}(-1)^n\dfrac{n}{n+1}\not = 0$ hence the series $\sum_{n\ge 0} (-1)^n\dfrac{n}{n+1}$ can't be convergent. It is then a divergent series.

Answer 2

Try the divergence test. If the limit $$ \lim_{n\to\infty}(-1)^n\frac{n}{n+1} $$ isn't $0$, then it diverges.

Answer 3

For series $\sum_{i=1}^{i=\infty}a_n$.if $\lim_{n\to\infty}a_n\neq0$,then series diverges