infinity - Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form?

--EDIT--

We can also write the sum for any constant $k$ from $0$ to $n$ as $k(n+1)$

So, $\displaystyle \sum_{n=0}^\infty 0 = 0 \times (\infty + 1)$ which is an IF.

Why does wolframalpha say that it is convergent?

Thank you,