1549901369-Elements_of_Real_Analysis__Denlinger_

502 Chapter 8 • Infinite Series of Real Numbers

n EN. Define a'= (la1l,la2I,··· ,lanl) and f! = (lbil,lb 2 I,··· ,lbnl). Then ,
00
using the Cauchy-Schwarz inequality, the partial sums of I:: lakbkl satisfy
k=l
n
L lakh I = la1 b1 I + la2b2 I + · · · + lanbn I
k=l
= (la1I, la2I, · · ·, lanl) · (lb1I, lb2I, · · ·, lbnl)

= a'· f! ~ ~~ (Cauchy-Schwarz inequality)

~ Jla11^2 + la 212 + · · · + lanl^2 )lbil^2 + Jb2l^2 + · · · + lbn l^2

n n

= I:: iakl^2 I:: lbkl^2

k=n l n k=l

=I:: a% I:: b%

k=l 00 k=l 00

~ I:: a% I:: b%.

k=l k=l

(These series converge since { ak} and {bk} are square

summable.)

Hence the sequence of partial sums {k~l lakbk I} is bounded above. So, by

00 00

Theorem 8.2.2, I:: lakbk l converges; i.e., I:: akbk converges absolutely. •

k=l k=l

Square summable sequences are of great importance in the study of "se-

quence spaces," esp ecially Hilbert Space.

EXERCISE SET 8.5

1. Find convergent series I:: ak and I:: bk such that I:: akbk diverges.


2. Find divergent series L ak and L bk such that L akbk converges.


3. Prove Theorem 8.5.1.^8


4. Use Theorem 8.5.1 to prove that if I:: ak and I:: bk are both absolutely
   convergent, then so is L akbk.


5. To see the similarity between Abel's "summation by parts" formula and
   the familiar integration by parts formula, work out the details of the
   following. Define ao = 0 and regard the sequence {an} as the sequence
   of partial sums of a series L Xk (see Exercise 8.1.6). Also, Vk EN, define
   6-ak = ak+l - ak and 6-bk = bk+1 - bk. With these definitions, using


6. This exercise is identical to Exercise 8.3. 12.