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I think it does but I'm having trouble showing it.

  • Using the integral test: the function $f(x)=\frac{1}{\ln(x^x+x^2)}$ is decreasing but I'm having trouble integrating $\int_{1}^{\infty} \frac{dx}{\ln(x^x+x^2)}$, trying the substitution $t=x^x$ didn't seem to work.

  • The root test gives $\sqrt[n]{\left |\frac{1}{\ln(n^n+n^2)}\right |} \to 1$ which doesn't imply anything.

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    You should be try$i$$n$g a limit compariso$n$. First show that you can ignore the n^2. (The way to use the integral test is generally not to integrate the function itself, but to integrate obvious lower and/or upper bounds on the function which are easier to integrate.)2011-01-10

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Hint: Compare with $\sum_{n=1}^{\infty} \frac{K}{n\log n}$

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    @daniel: Yes that would prove it. Of course, I have no clue if that is true :-)2011-01-10