could anyone help me with the limit of the following sequence :
$S_n=\sqrt[n(n+1)]{\prod _{k=0}^{n}\binom{n}{k}}$
Using the GM-AM inequality, we can prove that :
$S_n\le \frac{2}{(n+1)^{\frac{1}{n}}}\rightarrow2$ and I've been trying to find a sequence $u\le S$ that converges to $2$ (eventhough I have no reason to believe such a sequence exists !) but I can't find any.
Thank you in advance for your help !
Asked
Active
Viewed 82 times
6
Tengen
- 1,016
- 5
- 10
-
It seems from the graph that ${S_n}$ is increasing but the limit is less than $\mathrm e^{0.5}≈1.6487$. – Ѕᴀᴀᴅ Jun 01 '18 at 01:09