220 Chapter 7Sequences and series
and this can be rearranged as the Cauchy product
(7.32)
where
The radius of convergence of the product is equal to the smaller ofR
AandR
B.
EXAMPLE 7.16Multiplication of power series
0 Exercise 78
(iii) Differentiation and integration.
A power series may be differentiated or integrated term by term to give a power series
whose radius of convergence is the same as that of the original series. Thus, term by
term differentiation of
gives
(7.33)
Similarly, termwise integration of the seriesA(x) gives
(7.34)
The radius of convergence of both the derived series and integrated series isR 1 = 1 R
A.
=+ +cax + +
a
x
a
x
01 2 2 323
ZZA x dx a x dx
a
r
xc
rrrrr r() ==
==+∑∑
0011
∞∞dA
dx
ra x a a x a x
rrr==+++
=−∑
1112 3223
∞Ax ax a ax ax ax
rrr()==++++
=∑
001 2233∞=+ +
!
1 +=
2
2x
x
e
x=+ ×− +×
+×
−
!
+×− +
!
11 231 1 ×
2
2
32
3
2
22()
()
x () 11
2
x +
ee x
x
x
x
32 xx213
3
2
12
2
×=++
!
+− +
−
−()
()
(
))
22!
cab
ririri=
=−∑
0Cx ab ab ab x ab ab ab x()=+ + + ++( ) ( ) +
00 01 10 02 11 202==
=∑
rrrcx
0∞