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Evaluate: $$\mathop {\lim }\limits_{n \to \infty } \frac{{m(m - 1)....(m - n + 1)}}{{(n - 1)!}}{x^n}$$ where $x \in (-1,\ 1)$

I tried solving this as follows: $$\mathop {\lim }\limits_{n \to \infty } \frac{{mx}}{{n - 1}}\frac{{(m - 1)x}}{{n - 2}}.....\frac{{(m - n + 1)x}}{1}$$ $$=\{ \mathop {\lim }\limits_{n \to \infty } \frac{{mx}}{{n - 1}}\} \{ \mathop {\lim }\limits_{n \to \infty } \frac{{(m - 1)x}}{{n - 2}}\} .....\{ \mathop {\lim }\limits_{n \to \infty } \frac{{(m - n + 1)x}}{1}\} $$

Now as the individual limit $\ \mathop {\lim }\limits_{n \to \infty } \frac{{mx}}{{n - 1}}$ evaluates to $0$ can we say that the limit as whole tends to $0$?

I personally feel that this reasoning seems incorrect. Can someone please suggest an alternate way of tackling this problem.

Thanks!

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  • $\begingroup$ Not all the limits are $0$. For example the last one isn't. $\endgroup$ – user258700 Nov 26 '15 at 16:50
  • $\begingroup$ Yes, that's what I felt while solving this problem. In fact, the last limit is $ - \infty $. However, I wasn't able to think of any other way to proceed. $\endgroup$ – Kaustav Sen Nov 26 '15 at 16:52
  • $\begingroup$ Are you sure that you have typed in everything correctly? $\endgroup$ – user258700 Nov 26 '15 at 16:52
  • $\begingroup$ Yes, I just cross-checked it with the question given in the book $\endgroup$ – Kaustav Sen Nov 26 '15 at 16:54
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    $\begingroup$ What number is m? Natural or any real mumber? $\endgroup$ – kmitov Nov 26 '15 at 16:56
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Note that let $p_n = {m-n+1 \over n-1}x$ and $\lim_n p_n = -x$.

Take $|x|< \beta < 1$, then for some $N$ we have $|p_n| < \beta$ for all $n \ge N$.

Then $|p_2\cdots p_n | \le |p_2 \cdots p_{N}| \beta^{n-N} $, hence $\lim_n p_2\cdots p_n = 0$.

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If you know the gamma function:

$$\lim_{n\to\infty}\left(\frac{{m(m - 1)....(m - n + 1)}}{{(n - 1)!}}\cdot x^n\right)=\lim_{n\to\infty}\left(\frac{\prod_{k=1}^{n}\left(m-k+1\right)}{{(n - 1)!}}\cdot x^n\right)=$$ $$\lim_{n\to\infty}\left(\frac{\frac{m!}{(m-n)!}}{{(n - 1)!}}\cdot x^n\right)=\lim_{n\to\infty}\left(\frac{\frac{m!}{(m-n)!}}{{(n - 1)!}}\cdot x^n\right)=$$ $$\lim_{n\to\infty}\left(\frac{nm!}{n!(m-n)!}\cdot x^n\right)=\lim_{n\to\infty}\left(\frac{x^nm!}{\Gamma(n)\Gamma(m-n+1)}\right)=$$ $$m!\lim_{n\to\infty}\left(\frac{x^n}{\Gamma(n)\Gamma(m-n+1)}\right)=m!\lim_{n\to\infty}\left(\frac{x^n}{\Gamma(n)(m-n)!}\right)$$

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  • $\begingroup$ And what hapens then? $\endgroup$ – kmitov Nov 26 '15 at 21:18
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If $m$ is a positive integer the limit is evidently zero.

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  • $\begingroup$ Why the downvote? $\endgroup$ – copper.hat Nov 26 '15 at 16:59
  • $\begingroup$ I wanted to ask the same. $\endgroup$ – kmitov Nov 26 '15 at 16:59
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    $\begingroup$ Welcome to the club of victims of downvoters ! $\endgroup$ – Claude Leibovici Nov 26 '15 at 17:22
  • $\begingroup$ What do you meen? $\endgroup$ – kmitov Nov 26 '15 at 21:17
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Hint: Basically, you are asking us to evaluate $\displaystyle\lim_{n\to\infty}n~{m\choose n}~x^n$ for $m\not\in\mathbb Z$ and $|x|<1$. The obvious choice would be Stirling's approximation.

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