Multiplying infinite sums

Question

Is that true for infinite sums that $c\cdot \sum_{n=1}^{\infty} a_n=\sum_{n=1}^{\infty} c\cdot a_n$?Or it only applies when the sum is finite($\sum a_n \lt \infty$)?

Answer 1

Well it depends where you are considering the question... If we are in $\mathbb{R}$ or $\mathbb{C}$ then $\infty$ is not memeber of these sets so your equality have no sens. However if you note $S_n = \sum_{k=1}^n a_n$ and $T_n = \sum_{k=1}^n ca_n$ then it is true that $cS_n=T_n$ for every $n$ (as long as $|S_n|,|T_n| < \infty$) and you know that if $S_n$ is a convergent sequence then also is $\alpha S_n$ for any $\alpha \in \mathbb{R}$, thus $T_n$ is also convergent.