An Imporper Integral

Question

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.

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.
http://en.wikipedia.org/wiki/Term_test — Platatat, Feb 28 '14 at 16:53

score 0 · Answer 1 · answered Feb 28 '14 at 16:51

0

Hint

The integral is divergent since

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

answered Feb 28 '14 at 16:51

The funny thing is, that I just made the same conclusion! Thank you for your help! – Danny Feb 28 '14 at 16:52

An Imporper Integral

1 Answers1