# What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

#### Jack

##### New member
What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

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#### Sudharaka

##### Well-known member
MHB Math Helper
Re: What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?
Hi Jack, I suggest you should learn some LaTeX before posting questions since the characters that you use makes it difficult to understand what your question is. We have a nice collection of threads that you can use to learn LaTeX.

$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx$

Kind Regards,
Sudharaka.

#### Jack

##### New member
Re: What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

Hi Jack, I suggest you should learn some LaTeX before posting questions since the characters that you use makes it difficult to understand what your question is. We have a nice collection of threads that you can use to learn LaTeX.

$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx$

Kind Regards,
Sudharaka.
YEs。

#### Sudharaka

##### Well-known member
MHB Math Helper
Re: What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?

$\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx=2\int_{0}^{\infty}x^{n} e^{-nx}\,dx$

Substitute $$y=nx$$ and we get,

\begin{eqnarray}

\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx&=&\frac{2}{n^{n+1}}\int_{0}^{\infty}y^{n} e^{-y}\,dy\\

&=&\frac{2}{n^{n+1}}\Gamma(n+1)

\end{eqnarray}

I am assuming that $$n$$ is a positive integer. Then,

$\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx=\frac{2\Gamma(n+1)}{n^{n+1}}=\frac{2n!}{n^{n+1}}$

It could be shown that, $$\displaystyle\lim_{n\rightarrow \infty}\frac{2n!}{n^{n+1}}=0$$.

$\therefore \lim_{n\rightarrow \infty}\int_{-\infty}^{\infty}\left|x\right|^{n} e^{-n|x|}\,dx=\lim_{n\rightarrow \infty}\frac{2n!}{n^{n+1}}=0$