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

[Jack3]

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

What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?
Find the limit and prove your answer.

 

[Sudharaka]

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

What is lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗?
Find the limit and prove your answer.

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 http://www.mathhelpboards.com/f26/ that you can use to learn LaTeX.

Is this your integral?

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

Kind Regards,
Sudharaka.

 

[Jack3]

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 http://www.mathhelpboards.com/f26/ that you can use to learn LaTeX.

Is this your integral?

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

Kind Regards,
Sudharaka.

YEs。

 

[Sudharaka]

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, \(\displaystyle\lim_{n\rightarrow \infty}\frac{2n!}{n^{n+1}}=0\).

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

 

[blue_raver22]

The limit in question is a well-known integral in mathematics known as the Laplace transform. This integral is often used in the study of differential equations and has important applications in physics and engineering.

As n approaches infinity, the exponential term e^(-n|x|) will become smaller and smaller, approaching 0. This means that the integral will primarily be determined by the term x^n, which increases without bound as n approaches infinity.

To prove this, we can use the Dominated Convergence Theorem, which states that if a sequence of functions f_n converges pointwise to a function f and if there exists a function g such that g is integrable and |f_n(x)| ≤ g(x) for all x, then the integral of f_n converges to the integral of f.

In this case, we can choose g(x) = x^n, which is integrable over the entire real line. We can also see that |f_n(x)| ≤ g(x) for all x, since the exponential term will eventually become negligible as n approaches infinity. Therefore, by the Dominated Convergence Theorem, we can conclude that the limit of the integral is equal to the integral of the limit, which is just the integral of x^n.

As n approaches infinity, the integral of x^n will diverge, meaning that there is no finite limit for this integral. In other words, the limit of the original integral is equal to infinity.

In conclusion, lim_(n→∞) ∫_(-∞)^∞〖〖 x〗^n e^(-n|x|) dm 〗 is equal to infinity. This result is important in the study of Laplace transforms and has many practical applications in fields such as engineering and physics.