<s>[INST] Compute: $\lim_{n \to \infty } \left ( 1-\frac{2}{3} \right ) ^{\frac{3}{n}}\cdots \left ( 1-\frac{2}{n+2} \right ) ^{\frac{n+2}{n}}$ Help me please to compute the limit of:
$
\lim_{n \to \infty }  \left ( 1-\frac{2}{3} \right ) ^{\frac{3}{n}}\cdot  \left ( 1-\frac{2}{4} \right ) ^{\frac{4}{n}}\cdot  \left ( 1-\frac{2}{5} \right ) ^{\frac{5}{n}}\cdots \left ( 1-\frac{2}{n+2} \right ) ^{\frac{n+2}{n}} 
$
What I did:
$
L=\frac{a_{n+1}}{a_n} =   \frac{ \left ( 1-\frac{2}{n+3} \right ) ^{\frac{n+3}{n+1}}}{ \left ( 1-\frac{2}{n+2} \right ) ^{\frac{n+2}{n}}}=\frac{2/e}{2/e}=1
$
But it's not. Since $L=1$, I need use something else...
 [/INST] $(1-2/k)^k$ approaches $e^{-2}$, so most of the factors are near $e^{-2}$.
Then you take the $n$th root of them all, and there are $n$ factors, so the answer is $e^{-2}$
</s>