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 8.8.2Find the Taylor series for e^{x} sinx centered at x = 0.

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 e^{x} and the one for sinx converge for all
x.

Example 8.8.3Find the Taylor series for tanx centered at x = 0.

Lets suppose it has a Taylor series a_{0} + a_{1}x + a_{2}x^{2} +

Using the above, a_{0} = 0,a_{1}x = x so a_{1} = 1,

( (−1) )
0 2 +a2

x^{2} = 0 so a_{2} = 0.

( a1)
a3 − 2

x^{3} =

−1
3!

x^{3}
so a_{3}−

1
2

= −

1
6

so a_{3} =

1
3

. Clearly one can continue in this manner. Thus the first several
terms of the power series for tan are

1 3
tan x = x + 3x + ⋅⋅⋅.

You can go on calculating these terms and find the next two yielding

tan x = x + 1x3 + 2-x5 +-17x7 + ⋅⋅⋅
3 15 315

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

(x)

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 sinx by the power series for cosx just like they were
polynomials.