Question 9.T.12: (Dirichlet Test for Uniform Convergence) Let (un) and (vn) b...
(Dirichlet Test for Uniform Convergence)
Let (u_{n}) and (v_{n}) be two sequences of real valued functions on a set D ⊆ \mathbb{R} which satisfy the following conditions:
(i) The sequence (U_{n}) of partial sums of (u_{n}) is uniformly bounded, in the sense that there is a constant K such that
|U_{n}(x)| = \left|\sum\limits_{k=1}^{n}{u_{k(x)}} \right| ≤ K for all x ∈ D, n ∈ \mathbb{N}.
(ii) The sequence (v_{n}) is monotonically decreasing on D; that is, v_{n+1}(x) ≤ v_{n}(x) for all n ∈ \mathbb{N} and x ∈ D.
(iii) v_{n} \overset{u}{\rightarrow} 0 on D.
Then the series \sum{u_{n}v_{n}} is uniformly convergent on D.
Let S_{n} = \Sigma^{n}_{k=1} u_{k} v_{k}. From Lemma 9.1, we have
S_{n} = U_{n}v_{n+1} + \sum\limits_{k=1}^{n}{U_{k}(v_{k} − v_{k+1})}.
If n > m and x ∈ D, noting that v_{k}(x) − v_{k+1}(x) ≥ 0, we obtain
|S_{n}(x) − S_{m}(x)| ≤ K |v_{n+1}(x)| + K |v_{m+1}(x)|
+ K \sum\limits_{k=m+1}^{n}{[v_{k}(x) − v_{k+1}(x)]}
= K |v_{n+1}(x)| + K |v_{m+1}(x)| + K[v_{m+1}(x) − v_{n+1}(x)].
Properties (ii) and (iii) imply v_{k}(x) ≥ 0 for all k, hence
|S_{n}(x) − S_{m}(x)| ≤ 2Kv_{m+1}(x).
Since v_{n} → 0 uniformly on D, we see that (S_{n}) satisfies the Cauchy criterion for uniform convergence.