When is it possible to split an infinite sum? I know ordering of terms matters for conditionally convergent sums, so must the sums be absolutely convergent to be able to split like so:
$\sum_{n=0}^{\infty} f_n + \frac{1}{n^2} = \sum_{n=0}^{\infty} f_n + \sum_{n=0}^{\infty}\frac{1}{n^2}$

If $\sum_{n=1}^\infty a_n$ exists and $\sum_{n=1}^\infty b_n$ exists, then the sum $\sum_{n=1}^\infty (a_n + b_n)$ exists and is equal to $$\sum_{n=1}^\infty a_n +\sum_{n=1}^\infty b_n$$

You can easily show this by using two facts:


*

*$$\sum_{n=1}^\infty a_n = \lim_{N\to\infty}\sum_{n=1}^N a_n$$

*If the limit $\lim_{n\to\infty} a_n$ exists and the limit $\lim_{n\to\infty} b_n$ exists, then the limit of $\{a_n+b_n\}_{n\in\mathbb N}$ also exists and $$\lim_{n\to\infty} (a_n + b_n) = \lim_{n\to\infty} a_n + \lim_{n\to\infty} b_n$$

