Question 9.T.11: (Weierstrass M-Test) Let (fn) be a sequence of functions def...
(Weierstrass M-Test)
Let (f_{n}) be a sequence of functions defined on D, and suppose that there is a sequence of non-negative numbers (M_{n}) such that
|f_{n}(x)| ≤ M_{n} for all x ∈ D, n ∈ \mathbb{N}.
If the series \sum{M_{n}} converges, then both \sum{f_{n}} and \sum{|f_{n}|} converge uniformly on D.
Let ε > 0. Since \sum{M_{n}} is convergent, there is an N ∈ \mathbb{N} such that
n > m ≥ N ⇒ \sum\limits_{k=m+1}^{n}M_{k} < ε.
But
\left|\sum\limits_{k=m+1}^{n}f_{k}(x)\right| ≤ \sum\limits_{k=m+1}^{n} \left|f_{k}(x)\right| ≤ \sum\limits_{k=m+1}^{n} M_{n} for all x ∈ D.
Therefore
n > m ≥ N ⇒ \left|\sum\limits_{k=m+1}^{n}f_{k}(x)\right| ≤ \sum\limits_{k=m+1}^{n} \left|f_{k}(x)\right| < ε for all x ∈ D,
and, by Cauchy’s criterion, both the series \sum{f_{n}} and \sum{|f_{n}|} are uniformly convergent.