If $\sum_n x_n$ converge to $L$, what is $\lim_{n \to \infty} x_n$?

Question

If $\sum_n x_n$ converge to $L < \infty$, what is $\lim_{n \to \infty} x_n$? This is a question I have been asked, but I cannot answer it. Can someone explain it to me?

Hint: What are the limits of the sequences $(y_n)$ and $(z_n)$ defined by $$y_n=\sum_{k=0}^nx_k\qquad z_n=\sum_{k=0}^{n+1}x_k\ ?$$ — Did, Jan 24 '16 at 15:54

score 4 · Answer 1 · edited Jan 24 '16 at 20:42

4

Hint: $$\lim_{ n\to \infty} x_n=\lim _{ n\to \infty} \left (\sum_{i=1}^{n} x_i -\sum_{i=1}^{n-1}x_i \right )$$ So what can you say about limit of $x_n$?

edited Jan 24 '16 at 20:42

Mutantoe

708

answered Jan 24 '16 at 15:53

Arpit Kansal

10,268

score 0 · Answer 2 · answered Jan 24 '16 at 15:51

0

If $\sum_{n}x^n$ converges to a finite number $L$, then we must have $x_n \to 0$. Without $x_n \to 0$, the sequence of partial sums cannot converge.

answered Jan 24 '16 at 15:51

Ben Grossmann

225,327

Could you prove it? – Jan 24 '16 at 15:53
@J.G see my answer,for proof! – Arpit Kansal Jan 24 '16 at 15:58
Yes, thanks a lot! – Jan 24 '16 at 15:59
@J.G You're welcome! – Arpit Kansal Jan 24 '16 at 16:05

If $\sum_n x_n$ converge to $L$, what is $\lim_{n \to \infty} x_n$?

2 Answers2