TITLE: How can I raise a Taylor Series to a power?
QUESTION [5 upvotes]: I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the following conclusion:
$$(\sum_{n=0}^\infty c_nx^n)^p = \sum_{{n_1}=0}^\infty \sum_{{n_2}=0}^\infty \cdots \sum_{{n_p}=0}^\infty c_{n_1}x^{n_1} c_{n_2}x^{n_2} \cdots c_{n_p}x^{n_p}.$$
Now, for multiplying two different Taylor series, this was as far as I could get, but assuming that it is a single Taylor series (as is the case for raising one to an exponent), I believe that one has: 
$$c_{n_1}x^{n_1} = c_{n_2}x^{n_2} = \cdots = c_{n_p}x^{n_p}.$$
Following from this, one would have the result:
$$\left(\sum_{n=0}^\infty c_nx^n\right)^p = \sum_{n=0}^\infty \left(c_nx^n\right)^p.$$
Is my logic mathematically sound, and is this a proper result? I have tried Googling this, but no results have come up. Thank you for your help.

REPLY [4 votes]: Recently, I had a similar problem.
The generating function for Bell polynomials 
seems to give the answer immediately:
$$ \left( \sum_{i=1}^\infty a_i x^i \right)^n = \sum_{m=n}^\infty 
 B_{m,n}(a_1,a_2,\ldots, a_{m-n+1})x^m, $$
where $B_{m,n}$ are the partial Bell polynomials.