Under what conditions can I rewrite the general term of a function series as the sum of a term and the following one?
Like in the following example, the general term was replaced with one with even exponent $2k$ and another one with the folloing (odd) number as exponent $2k+1$.
$$\sum_{n=2}^{\infty}{\ln\left(1+\frac{(-1)^n}{n}\right)} =\sum_{k=1}^{\infty}\left[{\ln\left(1+\frac{1}{2k} \right)} + \ln\left(1 - \frac{1}{2k+1}\right) \right]$$
Does the previous equality hold in any case or under some conditions? In particular must I know in advance that the series $\sum_{n=2}^{\infty}{\ln\left(1+\frac{(-1)^n}{n}\right)}$ is absolutely convergent before writing that equality?