Question 9.T.13: (Abel’s Test for Uniform Convergence) Suppose the sequences ...
(Abel’s Test for Uniform Convergence)
Suppose the sequences (u_{n}) and (v_{n}) of real valued functions defined on D ⊆ \mathbb{R} satisfy the following conditions:
(i) \sum{u_{n}} converges uniformly on D.
(ii) The sequence (v_{n}) is uniformly bounded and monotonically decreasing on D.
Then the series \sum{u_{n}v_{n}} is uniformly convergent on D.
Suppose |v_{n}(x)| ≤ M for all n ∈ \mathbb{N}, x ∈ D. If n > m then, applying the equality (9.15) to the sequences \bar{u}_{k} = u_{k+m} and \bar{v}_{k} = v_{k+m}, we obtain
\sum\limits_{k=1}^{n−m}{\bar{u}_{k}\bar{v}_{k}} = \bar{U}_{n−m}\bar{v}_{n−m+1} + \sum\limits_{k=1}^{n−m}{\bar{U}_{k}(\bar{v}_{k} − \bar{v}_{k+1})}.
⇒ \sum\limits_{k=m+1}^{n}{u_{k}v_{k}} = v_{n+1} \sum\limits_{k=m+1}^{n}{u_{k}} + \sum\limits_{k=m+1}^{n}{(v_{k}− v_{k+1})} \sum\limits_{j=m+1}^{k+m}{u_{j}}.
Now let ε be an arbitrary positive number. Relying on condition (i), we can choose N such that
n > m ≥ N ⇒ \left|\sum\limits_{j=m+1}^{n}{u_{j}(x)} \right| < ε for all x ∈ D.
Therefore, if n > m ≥ N, then for all x ∈ D we have
\left|\sum\limits_{k=m+1}^{n}{u_{k}(x)v_{k}(x)}\right| < ε |v_{n+1}(x)| + ε \sum\limits_{k=m+1}^{n}{[v_{k}(x) − v_{k+1}(x)]}
= ε |v_{n+1}(x)| + εv_{m+1}(x) − εv_{n+1}(x)
≤ 3Mε.
By the Cauchy criterion, the series \sum{u_{n}v_{n}} converges uniformly on D.