Can $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$ be written as a power series in a neighbourhood of zero?
Solution 1:
The answer is yes. In particular, note that with a substitution we can rewrite $$ f(x) = 2^{-(x^2 + 1)} \int_0^\infty t^{x^2 + 1}e^{-t}\,dt = 2^{-(x^2 + 1)}\Gamma(x^2 + 2). $$