1549901369-Elements_of_Real_Analysis__Denlinger_

496 Chapter 8 • Infinite Series of Real Numbers

Therefore, by the Cauchy criterion for convergence of series (Theorem
n
8.1.11), I: akbk converges. •
k=l

APPLICATION TO TRIGONOMETRIC SERIES

We shall apply Dirichlet's test to "trigonometric" series of the form
00 00
I: ak sin kt and I: bk cos kt.
k=l k=l
n n
We first want to prove that their partial sums I: ak sin kt and I: bk cos kt are
k=l k=l
bounded. Toward that end, we begin by proving two trigonometric identities.

Lemma 8.5.4 For all real numbers t not an integral multiple of 2n, and
VnEN,

n cos 1 - cos (n + l) t

(a) '°"' u sin kt =^2 2. t^2 ' and

k=l sm 2

n sin ( n + l) t -sin 1

(b) I:: cos kt = ~ t^2.

k=l 2sm 2

Proof. (a) Let t be a real number, not an integral multiple of 2n, and
n
Vn E N, let Sn = I:; sin kt. Recall the trigonometric identity
k=l
sinasin,8 = ~[cos(a - ,8) - cos( a + ,8)].
n
Thus, (sin!) Sn = I:; sin! sin kt
k=l
n
= ~ I:: [cos (! - kt) - cos (! + kt)]
k=l
n
= ~ I:: [cos (kt -! ) - cos (kt +! ) ]
k=l
= ~ [cos (! ) - cos ( ¥) + cos ( ¥) -cos (-¥) + · · · + cos ( nt -! ) - cos ( nt +! ) J

= ~ [cos (! ) - cos c2ntl)t)].

cos!- cos(n+ ~)t

Therefore, Sn =. t ·

2sm 2

(b) Exercise 6. •