# Convergence of a series

## Homework Statement

Let $$x_{n}$$ be a monotone increasing sequence such that $$x_{n+1}-x_{n}< \frac{1}{n}.$$ Must $$x_{n}$$ converge ?

## Homework Equations

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

## 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 ?

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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:
vela
Staff Emeritus