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The Weierstrass M-test provides a sufficient condition for a series of functions to be uniformly convergent. It is often handy since it is rather simple.

Proposition:

Let fn : S → C be a sequence of functions. Let Mn = supx∈S|fn(x)|. If

Σ1 Mk

[all sums are taken over k] converges then

F(x) = Σ1 fk(x)

converges uniformly.

Proof:

The partial sums of Σ1 fk(x) form a Cauchy sequence for any x, so the series is pointwise convergent and F(x) is defined for x ∈ S.

supx∈S1n fk(x) - F(x)| = supx∈Sn+1 fk(x)| ≤ Σn+1 Mn → 0

as n → ∞. Therefore the sequence of partial sums converges uniformly to F.


If we want to we can replace C by any Banach space (i.e. a complete normed space) without changing the conclusion.

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