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Let the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } } $$ Does this sequence converge?

I can tell intuitively that $a_n$ is monotonically increasing.

Therefore, there are two possibilities:

Either the sequence goes to infinity or it is bounded and therefore, converges to a finite limit.

How can show it?

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Interesting Problem! –  Hawk Feb 24 at 12:05
    
Does Stirling's Aprroximation work here? –  Hawk Feb 24 at 12:11
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1 Answer

up vote 10 down vote accepted

Note that \begin{align} \log a_n&=\log \sqrt{1!\sqrt{2!\cdots\sqrt{n!}}}=\frac{1}{2}\log 1! +\frac{1}{4}\log 2!+\cdots+\frac{1}{2^n}\log n! \\ &=\sum_{k=1}^n \frac{\log (k!)}{2^k}=\sum_{k=1}^n\frac{1}{2^k}\sum_{j=1}^k\log j= \sum_{k=1}^n \log k \Big(\sum_{j=k}^n \frac{1}{2^j}\Big). \end{align} Therefore, the sequence $\log a_n$, which is increasing, converges to $$ \log a_n=\sum_{k=1}^n \log k \Big(\sum_{j=k}^n \frac{1}{2^j}\Big)\longrightarrow\sum_{k=1}^\infty\frac{\log k}{2^{k-1}}=b<\infty. $$ Convergence can be established using for example the ratio test.

Thus $$ a_n\to \mathrm{e}^b=\exp\left(\sum_{k=1}^\infty\frac{\log k}{2^{k-1}}\right)=\prod_{k=1}^\infty k^{2^{-k+1}}. $$

Note. I am wondering whether $\sum_{k=1}^\infty\frac{\log k}{2^{k-1}}$ can be expressed in terms of some known constants.

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