The Real And Complex Number Systems

9.12 Assume that g n+1 (x) ≤ g n (x) for each x in T and each n = 1, 2, ...,
and suppose that g n → 0 uniformly on T. Prove that ∑ (−1) n+1 g n (x) converges
uniformly on T.
Proof: It is clear by Dirichlet’s Test for uniform convergence.
9.13 Prove Abel’s test for uniform convergence: Let {g n } be a sequence
of real-valued functions such that g n+1 (x) ≤ g n (x) for each x in T and for
every n = 1, 2, ... If {g n } is uniformly bounded on T and if ∑ f n (x) converges
uniformly on T, then ∑ f n (x) g n (x) also converges uniformly on T.
Proof: Let F n (x) = ∑ n
k=1 f k (x) . Then
s n (x) =
n∑
f k (x) g k (x) = F n g 1 (x)+
k=1
and hence if n > m, we can write
n∑
(F n (x) − F k (x)) (g k+1 (x) − g k (x))
k=1
s n (x)−s m (x) = (F n (x) − F m (x)) g m+1 (x)+
n∑
k=m+1
Hence, if M is an uniform bound for {g n } , we have
|s n (x) − s m (x)| ≤ M |F n (x) − F m (x)| + 2M
(F n (x) − F k (x)) (g k+1 (x) − g k (x))
n∑
k=m+1
|F n (x) − F k (x)| . (*)
Since ∑ f n (x) converges uniformly on T, given ε > 0, there exists a positive
integer N such that as n > m ≥ N, we have
|F n (x) − F m (x)| < ε
M + 1
By (*) and (**), we have proved that as n > m ≥ N,
|s n (x) − s m (x)| < ε for all x ∈ T.
Hence, ∑ f n (x) g n (x) also converges uniformly on T.
for all x ∈ T (**)
Remark: In the proof, we establish the lemma as follows. We write it
as a reference.
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