$\displaystyle f= (-1)^n \frac{n}{n+1}$
If it converges find the limit and if it diverges justify why.
It definitely does not converge. $\displaystyle \displaystyle \begin{align*} \frac{n}{n + 1} \to 1 \end{align*}$ as $\displaystyle \displaystyle \begin{align*} n \to \infty \end{align*}$, so eventually the terms will oscillate between -1 and 1.
Does this help you?
$\displaystyle \lim_{n\to\infty}(-1)^n\frac{n}{n+1}$
$\displaystyle \rightarrow \lim_{n\to\infty}(-1)^n\frac{n/n}{n/n+1/n}$
$\displaystyle \rightarrow \lim_{n\to\infty}(-1)^n\frac{1}{1+0}$
$\displaystyle \rightarrow \lim_{n\to\infty}(-1)^n(1)$
Thanks for the replies. I understand the sequence definitely diverges.
Please note my brother asked me this question and it is not part of my coursework(and i have never done anything like this before), which is why i didn't make an attempt at a solution, however i am still interested in the answer.
His question asked to find an appropriate epsilon and n to justify why the series diverges.