Define what it means for a bounded function $f:[a, \infty) \rightarrow \mathbb{R}$ to be Riemann integrable.

Show that a monotonic function $f:[a, b] \rightarrow \mathbb{R}$ is Riemann integrable, where $-\infty<a<b<\infty$.

Prove that if $f:[1, \infty) \rightarrow \mathbb{R}$ is a decreasing function with $f(x) \rightarrow 0$ as $x \rightarrow \infty$, then $\sum_{n \geqslant 1} f(n)$ and $\int_{1}^{\infty} f(x) d x$ either both diverge or both converge.

Hence determine, for $\alpha \in \mathbb{R}$, when $\sum_{n \geqslant 1} n^{\alpha}$ converges.