8.10. MULTIPLICATION OF POWER SERIES 177
It fails to be analytic because it is not correctly given by its power series in any openset.
7. Euler seems to have done the following to find eix. He knew the power series for ex
and so substituted ix for x and this gave
1+ ix+(i2x2)/2!+(ix)3 /3!+ · · ·
= 1+ ix− x2/2!− ix3/3!+ · · ·
then he grouped the terms and got
1− x2
2!+
x4
4!+ · · ·+ i
(x− x3
3!+
x5
5!+ · · ·
)and then recognized this as cosx+ isinx. What is wrong with this kind of thing?Why is it that in this case there is absolutely no problem and this is a legitimateexplanation of Euler’s formula.
8. Find limx→+∞x
x+sin(3x) . Hint: It might be good to not use L’Hospital’s rule.
8.10 Multiplication of Power SeriesNext consider the problem of multiplying two power series.
Theorem 8.10.1 Let ∑∞n=0 an (x−a)n, ∑
∞n=0 bn (x−a)n be two power series which
have radius of convergence r1 and r2, both positive. Then(∞
∑n=0
an (x−a)n
)(∞
∑n=0
bn (x−a)n
)=
∞
∑n=0
(n
∑k=0
akbn−k
)(x−a)n
whenever |x−a|< r ≡min(r1,r2) .
Proof: By Theorem 8.1.3 both series converge absolutely if |x−a| < r. Therefore, byTheorem 5.5.6 (
∞
∑n=0
an (x−a)n
)(∞
∑n=0
bn (x−a)n
)=
∞
∑n=0
n
∑k=0
ak (x−a)k bn−k (x−a)n−k =∞
∑n=0
(n
∑k=0
akbn−k
)(x−a)n .
The significance of this theorem in terms of applications is that it states you can multiplypower series just as you would multiply polynomials and everything will be all right on thecommon interval of convergence. This is called the Cauchy product.
This theorem can be used to find Taylor series which would perhaps be hard to findwithout it. Here is an example.
Example 8.10.2 Find the Taylor series for ex sinx centered at x = 0.