Here's some power series everyone should know:
$1/1-x$
$e^x$
$\sin x$
$\cos x$
$\log (1+x)$
these ones are all most people know.

You can also do these cute ones:
($\sinh$ and $\cosh$ are cute)
$\arctan x$
$1/(1+x^2)$ and more...

interesting but much harder is
$\tan x$
