When are infinite series distributive?

Question

Say you we're working with a sequence of numbers $(a_i)$ When can we perform this operation:

$c \sum_{i=1}^{\infty} a_i = \sum_{i=1}^{\infty} (c a_i)$ ?

Answer 1

It's always true when the series $\sum_{i=1}^\infty a_i$ converges. When $\sum_{i=1}^\infty a_i$ diverges, so does $\sum_{i=1}^\infty (c a_i)$, unless $c=0$.

Answer 2

If you are playing around with series, you can always do it, and worry about convergence and correctness later.

That is why you (and Euler) can say $\sum_{n=0}^{\infty} 2^n = -1$, by putting $x=2$ in $\sum_{n=0}^{\infty} x^n = \frac1{1-x}$,

In this case, look up "Divergent Series" and have fun.