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### Limit Ratio of Gamma Functions [duplicate]

How can one prove that for any $x>1$: $$\lim\limits_{n\to \infty }\left(\frac{n^x \Gamma \left(n+1\right)}{\Gamma \left(x+n+1\right)}\right)=1$$ It is easy to show this for $x$ a natural number ...
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### Evaluate $\int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx$

$$\int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx$$ My colleague got this problem from his friend but he didn't know the answer so he asked my help. ...
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### Prove $\int_0^1 \frac{t^2-1}{(t^2+1)\log t}dt = 2\log\left( \frac{2\Gamma \left( \frac{5}{4}\right)}{\Gamma\left( \frac{3}{4}\right)}\right)$

I am trying to prove that $$\int_0^1 \frac{t^2-1}{(t^2+1)\log t}dt = 2\log\left( \frac{2\Gamma \left( \frac{5}{4}\right)}{\Gamma\left( \frac{3}{4}\right)}\right)$$ I know how to deal with integrals ...
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### A closed form for the infinite series $\sum_{n=1}^\infty (-1)^{n+1}\arctan \left( \frac 1 n \right)$

It is known that $$\sum_{n=1}^{\infty} \arctan \left(\frac{1}{n^{2}} \right) = \frac{\pi}{4}-\tan^{-1}\left(\frac{\tanh(\frac{\pi}{\sqrt{2}})}{\tan(\frac{\pi}{\sqrt{2}})}\right).$$ Can we also find ...
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### Prove the existence of limit of $x_{n+1}=x_n+\dfrac{x_n^2}{n^2}$

The problem is: Let $\{x_n\}$ be a sequence such that $0<x_1<1$ and $x_{n+1}=x_n+\dfrac{x_n^2}{n^2}$. Prove that there exists the limit of $\{x_n\}$. It is easy to show that $x_n$ is increasing,...
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### How to find this infinite product

How to find this infinite product ? $$\prod_{n=0}^\infty \left(1-\dfrac{2}{4(2n+1)^2+1}\right)$$ I try to use infinite product of $\cos{x}$ but it doesn't work. Thank you.
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### A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
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### $\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show?

$$\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$$ Anyone an idea on how to prove this?
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### challenging alternating infinite series involving $\ln$

I ran across an infinite series that is allegedly from a Chinese math contest. Evaluate: $\displaystyle\sum_{n=2}^{\infty}(-1)^{n}\ln\left(1-\frac{1}{n(n-1)}\right).$ I thought perhaps this ...
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Please help me to evaluate this integral $$\int_0^1\frac{\log(1+x)-\log(1-x)}{\left(1+\log^2x\right)x}\,dx$$ I tried a change of variable $x=\tanh z$, that transforms it into the form $$\int_0^\infty\... 4answers 283 views ### Evaluating -\int_0^1\frac{1-x}{(1-x+x^2)\log x}\,dx I was trying do variations of an integral representation for \log\frac{\pi}{2} due to Jonathan Sondow, when I am wondering about if it is possible to evaluate$$\int_0^1-\frac{1-x}{(1-x+x^2)\log ...
I'm sorry if this is a simple question, but this page on Wolfram Research states that it follows from Stirling's formula that: $$\frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta$$ for large $x$, ...
The series $$\sum_{n=0}^\infty {{-\frac {1} 2} \choose n} \frac{(-1)^n}{2n+1}$$ is an endpoint for the Maclaurin series for arcsin(x). (The other endpoint is just the negative of this one.) I played ...