Integrating $\int_{a}^{\infty} \frac{x^4e^x}{(e^x-1)^2}dx$ for large values of $a$

Question

I have to show that $$\int_{0}^{\infty}\frac{x^4e^x}{(e^x-1)^2}dx \approx \int_{0}^{a} \frac{x^4e^x}{(e^x-1)^2}dx$$ for large values of $a$. This can be done by showing that $$\int_{a}^{\infty}\frac{x^4e^x}{(e^x-1)^2}dx$$ is exponentially small.

However, I am having trouble finding a way to integrate $$\int_{a}^{\infty}\frac{x^4e^x}{(e^x-1)^2}dx$$ analytically.

I was given a suggestion to think about making an approximation to $x^4$ when integrating over a region where $x$ is large, but I'm completely lost.

Any suggestions is greatly appreciated.

Answer 1

We have that for $a>\ln(2)$, $$\int_{a}^{\infty}\frac{x^4e^x}{(e^x-1)^2}dx=\int_{a}^{\infty}\frac{x^4e^{-x/2}}{e^{x/2}(1-e^{-x})^2}dx< \frac{\int_{0}^{\infty} x^4 e^{-x/2}dx}{e^{a/2}(1-e^{-\ln(2)})^2}=\frac{2^2\cdot 2^5\cdot 4!}{e^{a/2}}$$ which implies that the LHS goes to zero exponentially as $a$ goes to $+\infty$.

Answer 2

For sufficiently large $x$, $$\frac{x^4e^x}{(e^x-1)^2}=\frac{x^4}{(e^\frac{x}{2}-e^\frac{-x}{2})^2}$$ $$=\frac{x^4}{(\sum_{n=1}^{\infty} \frac{x^{2n}}{2^{2n}(2n)!})^2} <\frac{x^4}{\sum_{n=1}^{\infty} \frac{x^{4n}}{(2^{2n}(2n)!)^2}}$$ $$<\frac{x^4}{\frac{x^8}{k}}$$ $$=\frac{k}{x^4}$$ where $k$ is some constant. Thus, $$0<\frac{x^4e^x}{(e^x-1)^2}<\frac{k}{x^4}$$ Integrating both sides gives the required answer.