TITLE: limit $\lim_{n \to \infty} \frac{(2n+1)(2n+2) n^n}{(n+1)^{n+2}}$?
QUESTION [5 upvotes]: Wolfram alpha says that this limit (arising from a ratio test to determine the radius of convergence of a series) should be $4/e$. How does it get this result?

REPLY [5 votes]: Recall that
$$\require{mathtools}\left(1+\frac1n\right)^n\xrightarrow[n\to\infty] \ e$$
We have
$$ \frac{(2n+1)(2n+2) n^n}{(n+1)(n+1)^{n+1}}=\frac{2(2n+1) }{n+1}\left(1+\frac1n\right)^{-n} \xrightarrow[n\to\infty] \ 4/e$$