During my study to gamma function I find this form but I try to prove but I can't $$\Gamma(x) = \lim_{n\to \infty} \frac {n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)} $$ So I want to know how I can prove this limit ?
121955
During my study I find a form for gamma function it was $\Gamma (x) = \lim_{n\to\infty} \frac{n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)}$ And then by simplify this limit I get  $$\lim_{n\to\infty} \frac{n^{x-1}}{{x+n-1\choose n}}$$      then I use Stirling's approximation and evaluate the limit to $\ (x-1)! * e^{x-1} * (\frac{n}{n+x-1})^{x+n-1{\over2}}$$ What next I should do to get Gamma function from this limit ?
75593
All I know of the Gamma function is $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$ and the recursive formula $$\Gamma(\alpha) = (\alpha-1)\Gamma(\alpha - 1)\text{.}$$ A friend asked me to try to find $\Gamma(2.7)$. I assume this involves complex analysis, which I have absolutely no background on (except for the basic Euler formula). Using Wolfram, I found that  $$\Gamma(2.7) = e^{0.434821}$$ which suggests to me perhaps writing out $e^{-x}$ as a Taylor series: $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}\sum\limits_{i=1}^{\infty}\dfrac{(-x)^{i}}{i!}\text{ d}x$$ but alas, this doesn't look helpful.  How does one derive that $\Gamma(2.7) = e^{0.434821}$?
96544
Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$.  The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$.  I tried to find explicit formula for $x_n$, but did not succeed. I stucked at that point: if $a_1=(\frac{1}{4})^2$, $a_n=(a_{n-1}+\frac{1}{4})^2$, then $x_n=-a_{n-2}+\frac{1}{4}$ for $n\ge3$.   Thanks for any ideas and help.
61993
What is the limit of:  $$\lim_{n\to \infty} \frac{n^1+\dots+n^n}{1^n+\dots+n^n} = ?$$  I did some computation with big numbers, I guess it is in the interval $\left(\frac{1}{2},1\right)$.
47565
I get that the limit is  $$\lim\limits_{n\to\infty}f(x)=1$$ because $$\lim\limits_{n\to\infty}(1+x^{2n})=1,\,\,\lvert x\rvert&lt;1.$$  Is this right?
115761
I am trying to find this limit:  $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$  I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you
32818
Using the  for $\Gamma(x)$ and $\Gamma\Big(x + \frac12\Big)$, how can I prove the ? This is problem $10.7.3$ in the book , by Boros and Moll.  Any help is highly appreciated.
36258
Let $0 &lt; x_1 &lt; 1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ for $n \geqslant 1$.      Prove that the limit exists and find the limit in terms of $x_1$.   I have proved the existence but cannot manage the other part.  Thanks for any help.
83745
If $f:\mathbb{R}\to\mathbb{R}$ be a differentiable at $x=c$ then how can I show that $$\lim_{n\to\infty}\left\{f(c+\frac{1}{n^2})+f(c+\frac{2}{n^2})+\cdots+f(c+\frac{n}{n^2})-nf(c)\right\}=\frac{1}{2} f'(c)$$ I had no clue which result should I use to prove this. Please give some hints.   Thanks!
21140
What is  the value of $\Gamma(\mathrm{i})$ ? $\Gamma(z)$ is Gamma function. Here $\mathrm{i}^2=-1$.Can you help me with this problem ?
57355
I need to prove that :  $$\lim_{n \to \infty} \frac{x^n}{n!}=0$$  I tried to prove it using epsilon-delta, but I still haven't managed to prove it.  I think that maybe with sandwich theorem it possible to prove it.  Any help will be appreciated.
18978
Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$
98641
How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$.  Please help me.  Thanks!
36808
I am trying to prove that   $\lim_{n\to\infty} (\frac{\sum^n_{k=1} \frac{1}{k}} { \ln(n) })^{ \ln(n) } = e^γ$ where $γ$ the Euler-Mascheroni constant. We know that that:  $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$  By approximating the sum with integrals $$ \ln(n+1)=\int_1^{n+1}\frac1x\mathrm dx\le\sum_{k=1}^n\frac1k\le 1+\int_1^n\frac1x\mathrm dx=1+\ln n $$ since $1/k$ is decreasing for $k\ge 1$. We have that $$ \frac{\ln(n+1)}{\ln n}=\frac{\ln n+\ln(1+\frac1n)}{\ln n}\to1 $$ as $n\to\infty$ and we obtain the result. But I can't calculate $\lim_{n\to\infty} (\frac{\sum^n_{k=1} \frac{1}{k}} { \ln(n) })^{ \ln(n) } = e^γ.$ Any help?
72860
I'm working on a practice exam for my masters quals and I am having difficulties with the following limit. According to wolfram alpha, it's value is 1/2. Does anyone know how to find this limit?   $$\lim_{n\to\infty} \left[ n-{n\over e}\left(1+{1\over n}\right)^n \right].$$
20697
I am trying to find the limit of:  $$\lim_{n \to \infty} n(\sqrt[n]{2} - 1)$$  I know it should be very simple but I don't seem to get it. Thanks in advance for any help!
32918
According to Wikipedia, the  is related to the gamma distribution by the following relation:  $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$  Can you point me to a derivation of this fact? Can it be generalized? For example, is there a similar relation that results in something other than a constant 1 for the Gamma second parameter? What if we have  $$\lim_{n\to\infty,m\to\infty,n=mb}n B(k, m) $$  That is, the two variables go to infinity while maintaining a constant ratio b.  The reason I'm asking is because I'm trying to figure out how to simplify a hieraerchical bayesian model involving the beta distribution.  (This is my first post; sorry for the math notation, the MathJaX syntax was too daunting, but I'll try to learn)