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I wanted to examine the convergence of the series $\displaystyle x_{n+1}= \frac{n}{n+1} x_n$ and try to find its limit, but I'm having difficulty doing so. The only test I thought would be useful (ratio test) was inconclusive and I'm having trouble proving $0 \leq x_n \leq 1$ that I think could help end up with a geometric sequence. Any suggestions are welcome.

$x_1 = 1$ (This is $\forall n \geq 1$)

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    Try finding the first few values $x_n$ takes, and try to spot the pattern2012-12-15

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After a little effort in filling the dots, you will see the following:

$x_{n+1} = \frac{n}{n+1} x_n = \frac{n}{n+1}\frac{n-1}{n}x_{n-1} = \cdots = \frac{x_1}{n+1} = \frac1{n+1}$

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    yes, I meant that the series did not converge, thank you.2012-12-16
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You're given a sequence such that

$x_1=1$

$x_{n+1}=\frac n {n+1} x_n$

Define $u_n=nx_n$

Then the sequence becomes

$u_1=1\cdot x_1=1$

$u_{n+1}=u_n$

It follows that $u_n=1$ for all $n$, so that

$x_n=\frac 1 n $

for each $n$. The limit of this series is, in turn, $0$.