11.6 Multiplication Of Power Series
Next consider the problem of multiplying two power series.
Theorem 11.6.1 Let ∑
n and ∑
n be two power
series having radii of convergence r1 and r2, both positive. Then
< r ≡
Proof: By Theorem 11.1.3 both series converge absolutely if
by Theorem 10.5.7
The significance of this theorem in terms of applications is that it states you can
multiply power series just as you would multiply polynomials and everything will be all
right on the common interval of convergence.
This theorem can be used to find Taylor series which would perhaps be hard to find
without it. Here is an example.
Example 11.6.2 Find the Taylor series for ex sinx centered at x = 0.
All that is required is to multiply
From the above theorem the result should be
You can continue this way and get the following to a few more terms.
I don’t see a pattern in these coefficients but I can go on generating them as long as
I want. (In practice this tends to not be very long.) I also know the resulting power series
will converge for all x because both the series for ex and the one for sinx converge for all
Example 11.6.3 Find the Taylor series for tanx centered at x = 0.
Lets suppose it has a Taylor series a0 + a1x + a2x2 +
Using the above, a0 = 0,a1x = x so a1 = 1,
= 0 so a2
Clearly one can continue in this manner.
Thus the first several terms of the power series for tan are
You can go on calculating these terms and find the next two yielding
This is a very significant technique because, as you see, there does not appear to be a
very simple pattern for the coefficients of the power series for tanx. Of course there are
some issues here about whether tanx even has a power series, but if it does, the above
must be it. In fact, tan
will have a power series valid on some interval centered at 0
and this becomes completely obvious when one uses methods from complex analysis but
it isn’t too obvious at this point. If you are interested in this issue, read the
last section of the chapter. Note also that what has been accomplished is to
divide the power series for sin
by the power series for cosx
just like they were