Question

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

Answer 1

The Taylor series is defined by:
$f(x) = \sum \frac{f^n (a)}{n!} (x-a)^n$

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:
$f^n (x) = e^x g^{n-1} (x) + e^x g^n (x) \\ g^0 (x) = sin x$

Do this successively to n = 6.
$f^1 (x) = e^x (sinx +cos x) \\ f^2 (x) = e^x (2cos x) \\ f^3 (x) = e^x(2cos x - 2sin x) \\ f^4 (x) = e^x (-4 sin x) \\ f^5 (x) = e^x(-4sinx -4cos x) \\ f^6(x) = e^x(-8cos x)$

Plug in x=0 and sub into taylor series:
$e^x sin x = x+x^2 +\frac{x^3}{3} -\frac{x^5}{30}-\frac{x^6}{90}...$

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