The Real And Complex Number Systems

So,
n
s n ∑ cos2k − 1x
j1
sin 2nx
2sinx .
n
∑ j1
s ∑ n
j sin 2jx
j1
n
2n sin x
sin nx sinn 1x
2n sin x sin x
≤ 1 → 0
2nsin x 2
which implies that the given series has C,1 sum 0.
8.37 Given a series ∑ a n ,let
Prove that:
(a) t n n 1s n − n n
n
s n ∑
k1
Proof: DefineS 0 0, and thus
n
t n ∑ ka k
k1
n
n
a k , t n ∑
k1
∑ ks k − s k−1
k1
n
∑
k1
n
n
ks k − ∑ ks k−1
k1
∑ ks k − ∑k 1s k
k1
n
∑
k1
n−1
k1
n
ks k − ∑
k1
n 1s n − ∑
k1
n 1s n − n n .
ka k , n 1 n
n ∑ s k .
k1
k 1s k n 1s n
(b) If ∑ a n is C,1 summable, then ∑ a n converges if, and only if, t n on as
n → .
Proof: Assume that ∑ a n converges. Then lim n→ s n exists, say its limit a. By(a),we
have
t nn n 1 n s n − n .
Then by Theorem 8.48, we also have lim n→ n a. Hence,
n
s k