Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 5 / a / ii

Table of contents
- Solution ii

Solution

The Weierstrass M-test states: if functions

f_{n} : E \to R

or

C

satisfy

∣ f_{n} (x) ∣ \leq M_{n} for every x \in E, M_{n} \geq 0, \sum_{n = 1}^{\infty} M_{n} < \infty,

then

\sum f_{n}

has uniform convergence and absolute convergence on

E

. In fact

\sum ∣ f_{n} ∣

also has uniform convergence, since every tail is bounded by the corresponding tail of

\sum M_{n}

:

x \in E sup n = N + 1 \sum \infty ∣ f_{n} (x) ∣ \leq n = N + 1 \sum \infty M_{n} ⟶ 0.

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