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 f_{n} : S → **C** be a sequence of functions. Let M_{n} = sup_{x∈S}|f_{n}(x)|. If

Σ_{1}^{∞} M_{k}

[all sums are taken over k] converges then

F(x) = Σ_{1}^{∞} f_{k}(x)

converges uniformly.

**Proof:**

The partial sums of Σ_{1}^{∞} f_{k}(x) form a Cauchy sequence for any x, so the series is pointwise convergent and F(x) is defined for x ∈ S.

sup_{x∈S} |Σ_{1}^{n} f_{k}(x) - F(x)| = sup_{x∈S} |Σ_{n+1}^{∞} f_{k}(x)| ≤ Σ_{n+1}^{∞} M_{n} → 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.