If the series $\sum_{k=0}^{\infty} a_k$ converges then $a_k$ converges to 0.

Question

How to prove this theorem?

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score 3 · Answer 1 · answered Nov 05 '13 at 16:27

3

Hint: you have to really understand the definition of "convergent series" to take advantage of the following:

$$S_n:=\sum_{k=1}^na_k\implies a_n=S_n-S_{n-1}\;\;\ldots\ldots$$

answered Nov 05 '13 at 16:27

DonAntonio

I may be late here but I don't get this: $a_n = \sum_{i=1}^n a_i - \sum_{i=1}^{n-1} a_i \to L - L = 0$ as $n \to \infty$. We know (define) $\sum_{i=1}^n a_i$ converges to $L$, but how does that imply $\sum_{i=1}^{n-1} a_i$ converges to $L$ as well? This is trivial but for me I don't know how to prove? – Tomas Jul 10 '24 at 15:05

1 Answers1