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I am to find out whether the following Improper Integral converges:

$$\int_2^\infty \frac{e^{x/4}}{x^3{ln}^5x}\,dx\quad$$

Things that I've tried: Comparison with $$\frac{1}{x^3{ln}^5x}$$ Or with:(Which is impossible since it's not a "Decreasing" function) $$\frac{e^{x/4}}{x^3}$$ Or: $$\frac{{1}}{x}$$ Thanks in advance.

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    $\begingroup$ What is the limit of your integrand at $\infty$? $\endgroup$ Feb 28, 2014 at 16:50
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    $\begingroup$ maybe this helps $x^3ln^5x<x^8$ $\endgroup$
    – OBDA
    Feb 28, 2014 at 16:50
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    $\begingroup$ Using the Nth term test, the limit of this integrand as x goes to infinity is infinity, because exponential functions grow much faster than polynomial or ln. en.wikipedia.org/wiki/Term_test $\endgroup$
    – Platatat
    Feb 28, 2014 at 16:53

1 Answer 1

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Hint

The integral is divergent since

$$ \frac{e^{x/4}}{x^3{\ln}^5x}\ge\frac 1 x\quad \text{for $x$ large enough}$$

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  • $\begingroup$ The funny thing is, that I just made the same conclusion! Thank you for your help! $\endgroup$
    – Danny
    Feb 28, 2014 at 16:52

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