# Thread: Evaluating Improper Integrals - possibly using Weierstrass M-Test

1. ## Evaluating Improper Integrals - possibly using Weierstrass M-Test

Hi All

Any help on this would again be greatly appreciated:

Thanks Mathshelpforum

2. If $\displaystyle 0<r<3$ and $\displaystyle |z|\leq r$ then:
$\displaystyle \left| \frac{e^{z/n}\cdot z^n}{3^n + 2} \right| \leq \frac{|e^{z/n}|\cdot |z^n|}{3^n} \leq \frac{e^{x/n} r^n}{3^n} \leq K \left( \frac{r}{3} \right)^n$
Where $\displaystyle x=\mbox{Re}(z)$ and $\displaystyle K$ is an upper bound for $\displaystyle e^{x/n}$.

But, $\displaystyle \sum_{n=1}^{\infty} K \left( \frac{r}{3} \right)^n < \infty$ since this is a geometric series. So by the Weierstrauss-M test this converges.

3. ## Evaluating Improper integrals

Thanks for the post PerfectHacker

Any ideas on how to do part (ii) of the question?

Many thanks again

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