## Example 8.3.8 (b): Power Series?

Is the series

*f(x) = 3*a power series? If so, list center, radius of convergence, and general term^{2n}x^{n}*a*._{n}Yes, this is also a power series centered at *c = 0*. The
general term here is

a_{n}= 3^{2n}= (3^{2})^{n}= 9^{n}

According to our formula the radius of convergence is:

r = lim sup | a_{n}/ a_{n+1}| = lim sup | 9^{n}/ 9^{n+1}| = 1/9

Thus the series converges for *|x| < 1/9*. To determine
*exactly* where the series converges we should check the endpoints of
this interval manually:

Forx=-1/9:3, which diverges.^{2n}x^{n}= 9^{n}(-1/9)^{n}= (-1)^{n}Forx=1/9:3, which diverges as well.^{2n}x^{n}= 9^{n}(1/9)^{n}= 1^{n}

3^{2n}x^{n}f(x) =^{1}/_{1-9x}