I need to show that $f=\frac{1}{\sqrt x}$ is Lebesgue integrable on [0,1]. My attempt: I need to show $\sum_{m=n}^\infty \frac{m}{n} \mu(E_m^{(n)})$ converges absolutely $ \forall n$.
$\mu(E_m^{(n)})=\frac{n^2}{m^2}-\frac{n^2}{(m+1)^2}$
Then I fix $n=n_0$ and have this as my sum: $n_0 \sum_{m=n_0}^\infty \frac{2m+1}{m(m+1)^2}$ but I dont know how to proceed from this point.
Could you help me please.