## Convergent sequence or not?

1. The problem statement, all variables and given/known data
Let (x_n) be a real sequence which satisfies |x_n - x_(n+1)| < (1/n) for all natural numbers n.

Does (x_n) necessarily converge? Prove or provide counterexample.

2. Relevant equations
Cauchy Criterion for sequences

3. The attempt at a solution
I figured at first that this would be easily solved by determining if this sequence was a Cauchy sequence since the difference between the terms decreases with each successive term, but you don't know that you can always find a point after which the terms x_n, x_m have a difference of less than an epsilon. Any suggestions?
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 Can you think of a series whose partials sums diverge while satisfying the requirement: |s_n - s_(n+1)| < (1/n)?
 Well I was thinking the sequence of partial sums of the sequence x_n = 1/n diverges (Harmonic Series). But I guess since |x_n + x_(n+1)|<(1/n) that won't work.

## Convergent sequence or not?

In the sequence

$$s_n = \sum_{i=1}^n \frac{1}{i}$$

what is the value of |$$s_n - s_{n+1}$$|?
 |s_n - s_(n+1)| = 1/(n+1), so are you saying that will serve as a counter argument? Because doesn't the sequence x_n = 1/n converge?

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 Quote by zebraman |s_n - s_(n+1)| = 1/(n+1), so are you saying that will serve as a counter argument? Because doesn't the sequence x_n = 1/n converge?
1/n converges. But sethric is suggesting using the sequence s_n, not 1/n. If you know the series 1/n diverges then you know the sequence of partial sums diverges. If you want a more easily expressed answer you might want to think about using an approximation to s_n. What is it?
 Sorry, I don't understand what you're asking.
 Recognitions: Homework Help Science Advisor I'm asking if you know that s_n is approximately equal to log(n) by an integral test.
 No, but is that important?
 Dick is correct, I was suggesting to use the sequence s_n. You have already shown: |s_n - s_(n+1)| = 1/(n+1) < 1/n You have also already said s_n diverges.

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