# Homework Help: Convergence of a series

1. Aug 16, 2010

### ╔(σ_σ)╝

1. The problem statement, all variables and given/known data
Let $$x_{n}$$ be a monotone increasing sequence such that $$x_{n+1}-x_{n}< \frac{1}{n}.$$ Must $$x_{n}$$ converge ?

2. Relevant equations

Instinctively, I think it converges since the terms "bunch" up as n increases.

3. The attempt at a solution
$$|x_{n+1}-L| \leq |x_{n+1}-x_{n}| + |x_{n}-L|$$.
$$|x_{n+1}-L|< \frac{1}{n}+ |x_{n}-L|$$ . But this doesn't tell me anything about convergence.

If I apply the triangle inequality continuosly all I can see is that
$$|x_{n+1}-L|< \sum\frac{1}{n}$$

What can you guys tell me ?

2. Aug 16, 2010

### ╔(σ_σ)╝

Great that makes sense. I had a feeling it need not converge but I was unsure. Does this work too...
Define a sequence such that $$x_{n}-x_{n-1}= \frac{1}{n}$$ such that the sequence has positive terms them
[tex] |x_{n} -L| =|1 + 1/2 + ...+ 1/n|[/tex ] which does not converge.
I suppose that also works correct?
Excuse my latex, typing from a phone is difficult.

Last edited: Aug 16, 2010
3. Aug 16, 2010

### vela

Staff Emeritus
Yes, that works.

4. Aug 16, 2010

### ╔(σ_σ)╝

Thanks a lot for the help.

5. Aug 17, 2010

### gomunkul51

So what have you decided, does it converges or diverges? :)