1549901369-Elements_of_Real_Analysis__Denlinger_

492 Chapter 8 11 Infinite Series of Real Numbers

Then, n ~no=} ltakBn+l-kl < t lan+I-kBkl

k=l k=l no n

= L lan+i-kBkl + L lan+l-kBkl

k=l k=no+I

no n

:::; ML lan+I-kl + 2 ~, L lan+I-kl

k=l k=no+I

n

:::; M L lakl + ~· (18)

k=n+I-no

(show, in Exercise 13)

Since L lak I converges, the Cauchy criterion guarantees that 3 n1 E N 3
n
n > m > n1 =} L lak I <
2
~. Then
k=m+I
n
n >no+ n 1 =} n >no, n > n 1 , and n + 1 - no> n1 =} L lakl <
2
~.
k=n+I-no
Therefore,

n ~no +n1 =} ltakBn+I-kl < c:,

k=l

which proves the desired claim. •

In the statement of the previous theorem we were careful to indicate that
the convergence of the Cauchy product of an absolutely convergent series and
a convergent series need not be absolute. For an example illustrating this sit-
uation, see Exercise 9. For an example showing that a Cauchy product can
converge even when both the series diverge, see Exercise 1. In 1826 the Nor-
wegian mathematician Niels Henrik Abel proved that if the Cauchy product
of two convergent series converges, its sum must be the product of their sums.
Abel's theorem makes no use of absolute convergence. We shall be able to give
an easy proof of this result after we study power series. (See Exercise 8.6.18.)

EXERCISE SET 8.4

1. Consider the series L ak = 1 + 2 + 2 + 2 + 2 + 2 + · · · and L bk =
   1 - 2 + 2 - 2 + 2 - 2 + · · ·. Show that, although both of these series
   diverge, their Cauchy product converges. Find the sum of their Cauchy
   product.
   00 00


2. Let L ck denote the Cauchy product series of L rk and L (-l)krk. Find
   k=O k=O
   and simplify the expression for Ck and use it to prove that the product