<s>[INST] Evaluate $\lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n}$ What's the value of $$\lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n}?$$
I've tried the AM-GM inequality with no luck.
Also tried to right the inequality as:
$$\left(\left(2-\frac{1}{2}\right)\left(2-\frac{1}{3}\right)\dots\left(2-\frac{1}{n+1}\right)\right)^\frac{1}{n}.$$
Would be happy to hear any hints/proofs.
 [/INST] Clearly the expression is $a_{n}^{1/n}$ where $a_{n} = \dfrac{(2n + 1)!}{n!(n + 1)!2^{n}}$ and hence $$\frac{a_{n + 1}}{a_{n}} = \frac{(2n + 3)!}{2^{n + 1}(n + 1)!(n + 2)!}\cdot\frac{2^{n}n!(n + 1)!}{(2n + 1)!} = \frac{(2n + 3)}{(n + 2)} \to 2$$ so desired limit is also $2$. We have used the following standard result:
If $\{a_{n}\}$ is a sequence of positive terms such that $a_{n + 1}/a_{n} \to L$ then $a_{n}^{1/n} \to L$.
One can also use Stirling's approximation for $n!$ but proof of Stirling's approximation is not easy.
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