Taking a const out of a series

Question

$\sum\limits_{n = 1}^\infty {2{a_n}} $ is a series.
is it right to say the following?

$$\mathop {\lim }\limits_{n \to \infty } {S_{n = }}\sum\limits_{n = 1}^\infty {2{a_n}} = 2\sum\limits_{n = 1}^\infty {{a_n}} $$

Answer 1

Yes this true since a series $\displaystyle \sum_n a_n$ is nothing but the partial sum $\displaystyle\left(\sum_{k=0}^n a_k\right)_n$ which is a sequence and we know (and we can prove it using the definition) that for a sequence $(u_n)_n$ we have $$\lim_{n\to\infty}\lambda u_n=\lambda\lim_{n\to\infty} u_n$$