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Math Help - Evaluating Improper Integrals - possibly using Weierstrass M-Test

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

    Hi All

    Any help on this would again be greatly appreciated:

    Evaluating Improper Integrals - possibly using Weierstrass M-Test-q3c.jpg

    Thanks Mathshelpforum
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  2. #2
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    If 0<r<3 and |z|\leq r then:
    \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 x=\mbox{Re}(z) and K is an upper bound for e^{x/n}.

    But, \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.
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  3. #3
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    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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