# Tagged Questions

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### A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series $$\mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$$ I have no idea how to ...
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### Limit of factorial how to continue

$$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{(n+1)!-n!}}\right)=\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$ How to continue? the answer is $0$ ... thank you ...
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### How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10}$$ Can ...
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### Proving a complex sum equals factorial

I have just stumbled across the equality that: $$\sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n} \binom{n}{j} = n!$$ How would I go about proving this equality? Also, what is the left hand side equal to if ...
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### Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
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### limits of sequences exponential and factorial

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
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### Gamma function question

Gamma function is also known as generalized factorial function . why does the term "generalized" have been used ? Again, why is the Gamma function called Euler's second integral?
### How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$
I'd like a hint to show that: $$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$ Thanks.