Consider a sequence $a_{n}$ with $a_{n}=(-1)^{n} (\frac{1}{2}-\frac{1}{n})$. Let $ b_{n}=\sum_{k=1}^{n} a_{k}$ for all $n\in\mathbb{N} $. Then find
$$\limsup\limits_{n\to\infty} b_{n}\ \ \text{and}\ \ \liminf\limits_{n\to\infty} b_{n}$$
Please give some hint.
First study the 'inner workings' of the following two series:
$\tag 1 \displaystyle{\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}}$
$\tag 2 \displaystyle{\sum_{k=1}^{\infty} \frac{(-1)^{k}}{2}}$
The first series is known as the alternating harmonic series.
The second series doesn't converge, but you can still compute the $\text{lim sup}$ and $\text{lim inf}$
on the partial sums.
You'll also need to know some general identities that can be used when working with sigma summation.