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Not sure how to really get started with this because it seems like a geometric series. Can some provide me a hint as to how I should approach this series? I started to use the ratio test but quickly noticed the difficulty I'd have with simplifying it algebraically.

Is the ratio test the only test I can use?

$\sum_{n=1}^{\infty}\frac{(2n+1)^n}{n^{2n}}$

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    $\begingroup$ Are you familiar with the root test? $\endgroup$ – Eric Nov 25 '13 at 19:36
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Let $$u_n=\frac{(2n+1)^n}{n^{2n}}=\left(\frac{2n+1}{n^2}\right)^n$$ then by the Cauchy test we have $$\lim_{n\to\infty}(u_n)^{1/n}=\lim_{n\to\infty}\frac{2n+1}{n^2}=0<1$$ so the series is convergent.

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  • $\begingroup$ I'm not familiar with the Cauchy test... but thanks! $\endgroup$ – hax0r_n_code Nov 25 '13 at 19:38
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Use limit comparison test by comparing against $$\sum_{n=1}^{\infty} \dfrac3{n^n}$$ Look here for more info on $\displaystyle \sum_{n=1}^{\infty} \dfrac1{n^n}$.

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