Paper 1, Section II, D

Analysis I | Part IA, 2010

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

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

Prove that if f:[1,)Rf:[1, \infty) \rightarrow \mathbb{R} is a decreasing function with f(x)0f(x) \rightarrow 0 as xx \rightarrow \infty, then n1f(n)\sum_{n \geqslant 1} f(n) and 1f(x)dx\int_{1}^{\infty} f(x) d x either both diverge or both converge.

Hence determine, for αR\alpha \in \mathbb{R}, when n1nα\sum_{n \geqslant 1} n^{\alpha} converges.

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