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Show and prove if the following series converges or diverges

$$\sum_{j=1}^\infty{ \frac{(1+(1/j))^{2j}}{e^j}}$$

"I tried the comparison test, the root test, and the ratio test, but got messed up..."

Thanks!

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Apply the $\,n$-th root test:

$$\sqrt[n]{\frac{\left(\frac{n+1}n\right)^{2n}}{e^n}}=\frac{\left(\frac{n+1}n\right)^2}{e}\xrightarrow[n\to\infty]{}\frac1e<1\implies\;\text{the series converges}$$

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    $\begingroup$ Helps a lot! Thanks! $\endgroup$ – mflowww Nov 18 '13 at 0:19
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$(1+1/j)^j \to e$ as $j \to \infty$.

Therefore $(1+1/j)^{2j} \to e^2$ as $j \to \infty$, so each term is about $e^2/e^j$ which is exponentially small, so it converges.

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HINT: Recall the definition of $e$ $$ e=\lim_{n \to \infty} \bigg(1+\frac{1}{n}\bigg)^n $$ Notice the numerator can be written as $$ \bigg(1+\frac{1}{j}\bigg)^{2j} $$ and that this is similar to the above definition of $e$. What does this imply about your summation?

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