Why is the infinite series $(n+2n+3n+...)$ not $0$ when $n \to 0$?

Question

I thought the $\lim_{n \to 0} n+2n+3n+... = 0+2\cdot0+3\cdot0+... = 0 $

But we can write the above as $\lim_{n \to 0} n (1+2+3+...) = 0 (1+2+3+...)$

But $1+2+3+...$ goes to infinity and $0$ times infinity is not defined.

Answer 1

Because limits and infinite sums generally can't be interchanged:

$$+\infty=\lim_{n\to 0}{+\infty}=\lim_{n\to 0}n\sum_{k=1}^{+\infty} k=\lim_{n\to 0}\sum_{k=1}^{+\infty} nk\neq \sum_{k=1}^{+\infty}\lim_{n\to 0}nk=\sum_{k=1}^{+\infty} 0= 0$$