Question: I need some help here. I can show that if the Cauchy condensation test holds, then I get two separate series, one which converges by the comparison test, and one that converges by the ratio test. But I don't even know if this is a valid argument, since I'm not sure how to even check that the terms are decreasing. So I don't think this approach works.

I see that the same question has been asked [here](https://math.stackexchange.com/questions/1696221/proving-convergence-of-sum-n-1-infty-frac1n-log1-frac1n/), but I'm not really satisfied with the answers. Are there simple ways to determine convergence, with something like the comparison test?

Answer: 
**Hint:** For differential equation of form $f''=F(f,f')$, the standard track is set $v=f'$, then we have
$$
f''=\frac{\text dv}{\text dx}=\frac{\text dv}{\text df}\frac{\text df}{\text dx}=v\frac{\text dv}{\text df},
$$

then we get
$$
v\frac{\text dv}{\text df}=f^{-2}.
$$
$$y''=\frac 1 {y^2}$$
Multiply by $2y'$
$$2 y''y'=2\frac {y'}{y^2}$$

$$(y'^2)'=-2(y^{-1})'$$
Then integrate
$$y'^2=K-2y^{-1}$$