<s>[INST] Convergence of $\sum _{n=2}^{\infty }\:\frac{1}{n\ln\left(n\right)}$ Is this convergent or not?
$$\sum _{n=2}^{\infty }\:\frac{1}{n\ln\left(n\right)}$$
I tried using the ratio test but the limit is giving me 1, which doesn't help me. I don't think I'm supposed to use the integral test, since we haven't studied it.
 [/INST] If you are familiar with Cauchy Condensation Test, the condens series becomes 
$$\sum_{n} 2^n \frac{1}{2^n \ln (2^n)}$$
If not, you can use the idea of the test:
$$\frac{1}{2 \ln2 } \geq \frac{1}{2 \ln 2 }  \\
\frac{1}{3 \ln 3 } \geq \frac{1}{4 \ln 2^2 }  \\
\frac{1}{4 \ln 4 } \geq \frac{1}{4 \ln 2^2 }  \\
\frac{1}{5 \ln 5 } \geq \frac{1}{8 \ln 2^3 }  \\
.....\\
\frac{1}{2^n \ln 2^n } \geq \frac{1}{2^n \ln 2^n } $$
By adding you get
$$s_{2^n} \geq \frac{1}{2 \ln 2 } +\frac{1}{2 \ln 2^2 } +...+\frac{1}{2 \ln 2^n }=\frac{1}{2 \ln 2} (1+\frac12+\frac13+...+\frac1n)$$ 
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