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∈S |Σ1n fk(x) - F(x)| = supx∈S |Σn+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.