## Example 8.3.8 (e): Power Series?

Find center and radius of convergence for the power series

*n! (x + 3)*^{n}^{n!}/_{nn}(x - 4)^{n}

The first series has center *c = -3* and general term
*a _{n} = n!*. Therefore the radius of convergence is:

r = lim sup | a_{n}/ a_{n+1}| = lim sup^{n!}/_{(n+1)!}= lim sup^{n!}/_{(n+1) n!}= lim sup^{1}/_{n+1}= 0

That means the series only converges for *x = -3*, for no
other *x*.

The second series has center *c = 4* and general term
*a _{n} = ^{n!}/_{nn}*.
Therefore the radius of convergence is:

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

where we used our result on Euler's sequence.
That means the series converges for *|x - 4| < e*. To be picky
we need to investigate convergence at the endpoints 'manually', which is
left to the reader (sorry).