# prove/disprove absolute convergent

$\int_0^\infty 3^{-x}x^4cos(2x)dx$

I succeeded to prove that this integral is conditionally convergent with Dirichlet's test. I don't know how to prove/disprove absolutely convergent..

Thanks !

• What do you mean by " uniform convergence" of the integral ???? – Fred Aug 28 '17 at 9:08
• You right, fixed it. Thanks ! – Jill Aug 28 '17 at 9:20

You have that $$\lim_{x\to \infty }x^23^{-x}x^4\cos(2x)=0,$$ and thus $$3^{-x}x^4\cos(2x)=\mathcal O\left(\frac{1}{x^2}\right),$$ at the neighborhood of $+\infty$. Therefore it's absolutely integrable on $[1,+\infty )$. The integrability on $[0,1]$ is obvious. The claim follow.
• Because $$0=\lim_{x\to \infty }x^2 3^{-x}x^4\cos(2x)=\lim_{x\to \infty }\frac{3^{-x}x^4\cos(2x)}{\frac{1}{x^2}}$$ and thus $3^{-x}x^4\cos(2x)=o(1/x^2)$ which is in particular a $\mathcal O(1/x^2)$. @Jill – Surb Aug 28 '17 at 10:07