Find a power series representation of e^x sin(x)

The Taylor series is defined by:


Let a = 0.
Then its just a matter of finding derivatives and determining how many terms is needed for the series.

Derivatives can be found using product rule:


Do this successively to n = 6.


Plug in x=0 and sub into taylor series:


If more terms are needed simply continue the recursive derivative formula and add to taylor series.