Series with $\lim \sup$ and $\lim \inf$ being $+$ and $-\infty$

Question

Do you have an example of a series whose partial sums' $\lim \sup = \infty$ and whose partial sums' $\lim \inf = -\infty$ ?

I feel like it's like repeatingly adding a whole bunch of positive terms one time then substracting an even bigger whole bunch of negative terms, isn't that a way to get it ?

Answer 1

Let $f_{n}= n (-1)^{n}$ and the series be $\sum_{k=1}^{n} (f_{k}-f_{k-1})=f_n$.