8.4 The Cauchy Product of Series 493series converges when lrl < 1, and find its sum. Verify the conclusion of
Theorem 8.4.3 in this case.
CXl- Let lrl < l. Although the series L krk is not a geometric series, we can
k=O
find its sum by considering the Cauchy product of the geometric series
CXl
L rk with itself. Use the Cauchy product formula to write a power series
k=O
for ( f rk)
2
, thus getting a power series for (1
) 2. Massage this
k=O 1 - r
r
formula to show that the desired sum is (l _ r) 2- Use the result of Exercise 3 to find the sum of the given series.
CXl k 3 4 5
(a) I:-. (b) 1+1 + - + - + - + ...
k=l 5k 22 23 24 - Prove that if lrl < 1, f (k + l)(k +
2
) rk = (
1
) 3.
k=O 2 1 - r - Define the function E(x) = k~o ~~. This series converges absolutely, for
all real numbers x.^5 Use the Cauchy product of series to prove that
\:/x, y E IR, E(x)E(y) = E(x + y).
CXl (-l)k CXl 1
- Let L Ck denote the Cauchy product of the series L -k-and L k.
k=l k=l 3
Find and simplify the expression for Ck. Use one of the theorems of this
section to prove that this Cauchy product series converges and find its
sum. - Suppose {an} and {bn} are nonnegative sequences. Prove that the Cauchy
product of the alternating series I:(-l)k+^1 ak and I:(-l)k+^1 bk is also an
alternating series, and can be obtained by inserting alternating signs in
the Cauchy product of L ak and L bk.
CXl (-l)k+l CXl (-l)k+l
- Consider^6 the series L k 312 and L k 112. Show that one of these
k=l k=l
series converges absolutely and the other converges conditionally. Show
that the kth term of their Cauchy product series is
k
- ( )k+l'""" 1
Ck - -1 ~ i3 / 2(k + 1 - i)l/2.
- See Exercise 8.3.11.
- This example may be found in Burrill and Knudsen [25], page 143.