(i) State the Monotone Convergence Theorem and explain briefly how to prove it.

(ii) For which real values of $\alpha$ is $x^{-\alpha} \log x \in L^{1}((1, \infty))$ ?

Let $p>0$. Using the Monotone Convergence Theorem and the identity

$$\frac{1}{x^{p}(x-1)}=\sum_{n=0}^{\infty} \frac{1}{x^{p+n+1}}$$

prove carefully that

$$\int_{1}^{\infty} \frac{\log x}{x^{p}(x-1)} d x=\sum_{n=0}^{\infty} \frac{1}{(n+p)^{2}}$$