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When is it allowed to take a constant out of a series?

Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$?

It's pretty obvious when we know beforehand the series converges but what if we're asked to check if the series converges?

Can we take it out and then if it does converge then the operation was legal (like with LHR)?

Lastly, does it matter for diverging series?

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Remember that $ \sum_{n=1}^{\infty} a_n $ is defined by the limit of the sequence $ \sum_{n=1}^N a_n $ (when $ N\to \infty $). So by definition $$\sum_{n=1}^\infty c a_n =\lim_{N\to \infty} \sum_{n=1}^N c a_n = \lim_{N\to \infty} c \sum_{n=1}^N a_n = c \lim_{N\to \infty} \sum_{n=1}^N a_n = c\sum_{n=1}^\infty a_n $$ For every $c\neq 0$ if one limit exists so do the rest, so not only it doesn't change convergence/divergence of the series, if the series converges it's a true equality between $ \sum c a_n $ and $ c \sum a_n $.

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This is one of those few cases where assuming $c\neq 0$ then $\sum_{n=1}^\infty ca_n$ converges iff $\sum_{n=1}^\infty a_n$ converges. This follows from the definition of an infinite sum as the limit of partial sums and the limit rule for multiplication by a constant. From that we also easily get that $\sum_{n=1}^\infty ca_n=c\sum_{n=1}^\infty a_n$.

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