Adding two convergent series

Question

If $\sum_{n=1}^{\infty} a_n$ is finite and $\sum_{n=1}^{\infty} b_n$ is also finite, why is it that you can add the two series term by term and get the sum of the two series? Surely this is reordering the series.

Answer 1

The $N$:th partial sum $$ \sum_{n=1}^N (a_n+b_n) = \sum_{n=1}^N a_n + \sum_{n=1}^N b_n. $$ Hence, as $N \to \infty$, \begin{align} \lim_{N\to\infty} \sum_{n=1}^N (a_n+b_n) &= \lim_{N\to\infty} \left(\sum_{n=1}^N a_n + \sum_{n=1}^N b_n \right) \\ &= \lim_{N\to\infty} \sum_{n=1}^N a_n + \lim_{N\to\infty} \sum_{n=1}^N b_n \end{align} since both limits exist by assumption.

Answer 2

If $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ converge, $$\lim_{n\to\infty } (x_n+y_n)=\lim_{n\to\infty }x_n+\lim_{n\to\infty }y_n.$$

Set $x_n=\sum_{k=1}^n a_k$ and $y_{n}=\sum_{k=1}^n b_k$ and you'll get your result.