real analysis - Why this :$sumlimits_{n=0}^{infty}(0)=0$?

I'm confused how :$\sum\limits_{n=0}^{\infty} 0=0$ but after evaluation we w'd get $0.\infty$ which it's indeterminate case and wolfram alpha show here that is a convergent series which it's equal's $0 $, i'm only sure that is $0$ for it partial sum .

My question here is : Is there somone who show me how do I convince my self by :$\sum\limits_{n=0}^{\infty} 0=0$ ?

Thank you for any help

Answer

By definition, one has
$$

\sum_{n=0}^\infty a_n = \lim_{N \to \infty}\sum_{n=0}^N a_n
$$ giving
$$
\sum_{n=0}^\infty 0 = \lim_{N \to \infty}\sum_{n=0}^N 0= \lim_{N \to \infty}0=0.
$$