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$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that $S$ is finite and find an n so large that $S_n$ approximate $S$ to three decimal places.

Solution: first of all, I think that we Will use L'hopital rule and then use root test while starting to solve this. But how?

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Hint: use the $k$th root test.


If $a_k$ are the coefficients, that is:

$a_k=\left(\frac{k}{k+1}\right)^{k^2}$

then:

$\sqrt[k]{a_k}=\left(\frac{k}{k+1}\right)^{k}=\left(1-\frac{1}{k+1}\right)^k$

With some fantasy, do you recognize this sequence? Does it converge to something you know?

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    @rlgordonma: I didn't want to completely give it away, but yes, that is closer to an answer :).2012-12-22