1549901369-Elements_of_Real_Analysis__Denlinger_

8.5 Series of Products 497

n

Lemma 8.5.5 (a) Vt E JR, the partial sums I:: sin kt are bounded.

k=l

n

(b) Vt f=. 2p7r, (p E Z), the partial sums I:: cos kt are bounded.

k=l

n

Proof. (a) Lett be a real number and Vn E N , let Sn = I:: sin kt. If

k=l

n

t = 2p7r for some p E Z, all terms of I:: sin kt are 0, hence t his sum is bounded.

k=l

So in the remainder of the proof we assume t is not an integral multiple of 271".

cos ~ - cos ( n + ~) t

By Lemma 8.5.4, Sn =. t , so Vn E N,

2sm 2

IS I < icos(~)I + icos (n +~)ti < 2.

n - I 2 sin ~ I - I 2 sin ~ I

Since tis a fixed real number, ISnl is bounded above.

(b) Exercise 7. •

With the help of the two previous lemmas and Dirichlet's test, we can easily
prove the following important result about trigonometric series.

Theorem 8.5.6 If { ak} is a monotone decreasing sequence converging to 0,
then
00
(a) I:: ak sin kt converges Vt E JR, and
k=l
00
(b) I:: ak cos kt converges Vt f=. 2p7r, (p E Z)
k=l
(and may also converge when t = 2p7r).

Proof. Exercise 8. •

00 1
Examples 8.5.7 (a) I:: -k sin kt converges Vt ER
k=l
00 1
(b) I:: -cos kt converges Vt f=. 2p7r, and diverges Vt = 2p7r, (p E Z).
k=l k
00 1 00 1
( c) Both I:: 2 sin kt and I:: k 2 cos kt converge (absolutely) Vt E R
k=l k k=l

Another test of convergence of series of products, closely related to Dirich-
let's test, is Abel's test.