Associative property for series

Question

Are those equation always valid:
$$\sum a_n + b_n = \sum a_n + \sum b_n$$ $$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}$$

Answer 1

Those series must satisfy some conditions to do those things.

For instance, $\sum a_n$ and $\sum b_n$ must be convergent.

If you take $a_n = 1$ and $b_n = -1$ for all $n$, then $\sum a_n + b_n = 0$ but $\sum a_n$ and $\sum b_n$ are divergent.

I don't think you require more conditions.

The second equation is equivalent to the first one.

Answer 2

We can write:

$$\sum (a_n+b_n)=\sum a_n+\sum b_n$$

only if we know that either $\sum a_n$ or $\sum b_n$ converges.