definite integral/ limit of integral

**234578** · March 21st 2010, 11:34 PM

I need to find the definite integral from 1 to A of f(x)=1/[(x^(1+1/x)] in order to find the limit of this integral as A approaches infinity. I'm not sure if this is analysis or calculus but I just put it here because more than anything I need help finding the integral itself.

**shawsend** · March 22nd 2010, 04:00 AM

I believe it diverges because the function $f(x)=\frac{1}{x^{1+\frac{1}{x}}}$ is asymptotic to $g(x)=\frac{1}{x}$ as $x\to\infty$ since:

$\lim_{x\to\infty} \frac{f(x)}{g(x)}=1$

and therefore the convergence of $\int_1^{\infty} f(x)dx$ is the same as the convergence of $\int_1^{\infty}1/x dx$

Here's a nice reference on checking the convergence of improper integrals:

http://www.math.princeton.edu/~nelson/104/ImproperIntegrals.pdf

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