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Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.

I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.

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    did you try the divergence test?2012-11-09
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    @JavaMan Does this hint help? That sequence goes to $0$ as $n$ tends to infinity.2012-11-09
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    @F'OlaYinka - Are you sure? It doesn't seem to me that your inequality holds for $n\geq{3}$. You want to say that $\frac{2}{3}\leq{\frac{1}{2}}$?2012-11-09
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    @B.D. Thanks for the comment. I misread the sequence as $a_n = \left( \frac{n+1}{n}\right)^{n^2}$ for some reason.2012-11-09
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    @ Ittay Weiss: the Divergence Test fails since the expression tends to $0$.2012-11-09

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Hint: Try with Cauchy's root test.

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    $\displaystyle \left(1+\frac1n\right)^n$ does *not* approach $1$ as $n\to\infty$. For one thing, directly expanding it gives that $\displaystyle \left(1+\frac1n\right)^n\ge 1+n\left(\frac1n\right)=2$.2012-11-09
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    @ Seeker : The limit is $\frac 1e $ $<1$ , so Cauchy's root test in fact proves the covergence2012-11-09
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    Ah, lesson learned. Thanks all!2012-11-09
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Let $u_n$ = $ \Big ( \frac {n}{n+1} \Big)^{n^2}$ , then since $ \ \lim_{n \to \infty}\sqrt[n]{({\frac{n}{n+1}})^{n^2}} = \lim_{n \to \infty}(\frac{n}{n+1})^n = \lim_{n \to \infty}(\frac{1}{1+\frac{1}{n}})^n$ = $\frac1e$ $ < 1 $ , so by Cauchy's root test the series is convergent .

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Note the following: $$\Big(\frac{n}{n+1}\Big)^{n}=\Big(\frac{m-1}{m}\Big)^{m-1}=(1-1/m)^{m-1}$$ That formula/function seems familiar, doesn't it?

Thereafter, note that: $$\Big(\frac{n}{n+1}\Big)^{n^{2}}={\Big(\Big(\frac{n}{n+1}\Big)^{n}\space\Big)}^{n}$$ Hint: Simple comparison test with a sequence you know well.

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    Oh, snap - that was the "trick" I was looking for. Thanks, Much appreciated.2012-11-09
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    Nah, you could have used the root method. I just feel this is a lot more elegant.2012-11-09