Originally Posted by

**arbolis** It has probably been answered before on MHF, but I didn't find it using the search button.

Tell whether the following infinite series is convergent or not :

$\displaystyle \sum_{n=2}^{+\infty} \frac{1}{n \ln (n)}$. At first, I used the ration test, but I cannot conclude since it says "1". Then the integral test, which gave me divergent, but I'm not sure I did it well, because when I tried to get an approximation via Mathematica, I got an error like if it converges, it does it too slow or something like that. Nevertheless, it said an approximation is 13.4391. From it I doubted my integral test was well made. I tried the root test, but I'm not sure how to finish it.

Integral test : $\displaystyle \int_2^{+\infty} \frac{1}{x\ln (x)}$. Let $\displaystyle u$ be $\displaystyle \ln (x)$, then $\displaystyle du=\frac{1}{x}$, and the integral becomes $\displaystyle \int_2^{+\infty} \frac{du}{u}=\ln u \big | ^{+\infty}_2$. Replacing $\displaystyle u$ by $\displaystyle \ln (u)$, it's easy to see the integral diverges.

So normally, so does the infinite series. Am I right?