Question

Is this statement true?

I will put it in the comments.

Answer 1

If \[\left\{ a_{n} \right\}_{n=1}^{\infty}\] converges, then it is guaranteed that \[\sum_{n=0}^{\infty}a_{n}\] converges?

Answer 2

Yes, a series necessarily converges if its sequences of partial sums, ie the sequence you describe, converges. More formal language is available here: http://en.wikipedia.org/wiki/Convergent_series