## Example 8.3.8 (d): Power Series?

Is the series

*f(x) =*a power series? If so, what is the center and radius of convergence? List the first five coefficients^{(-1)n}/_{n!}(x + 1)^{2n}*a*._{0}, a_{1}, a_{2}, a_{3}, a_{4}If we write out the series we see it is indeed a power series with center
*c=-1* and all odd coefficients equal to zero:

f(x) = 1 - 1/1!(x+1)^{2}+ 1/2! (x+1)^{4}- 1/3! (x+1)^{6}+ ...

In particular:

- a
_{0}= 1/0! = 1- a
_{1}= 0- a
_{2}= 1/1! = 1- a
_{3}= 0- a
_{4}= 1/2! = 1/2

To find the radius of convergence we could apply our usual formula, but it
is a little confusing since every other *a _{n}* is zero.
Thus, we go the alternate route, as discussed in the text, to simply apply
the ratio test to the series. In other words, the series converges for
those

*x*for which:

But that means it converges for *all* *x*, or in other words
the radius of convergence *r = *.

Incidentally, this series also has a simpler expression, as you can see
from the plots below. After finishing the chapter, make sure to return here
to figure out *which* function this series represents.

^{(-1)n}/_{n!}(x + 1)^{2n}f(x) = mystery