I was playing around with Taylor series and a shifted Gamma Function and found something that doesn't work, and I'm not sure what I'm doing wrong. Is it some uniform continuity problem?
Let: $$\Pi(n)=\int_0^\infty t^ne^{-t}dt$$
For non negative integers $n$, this is $n!$.
Suppose we have a Maclaurin Series for some $f(t)=\sum_{k=0}^\infty c_kt^k$ where $c_k=\frac{1}{k!}\frac{d^kf(t)}{dt^k}|_{t=0}$
Then I'd expect:
$$\int_0^\infty f(t)e^{-t}dt=\int_0^\infty\sum_{k=0}^\infty c_kt^ke^{-t}dt=\sum_{k=0}^\infty\int_0^\infty c_kt^ke^{-t}dt=\sum_{k=0}^\infty c_kk!=\sum_{k=0}^\infty\frac{d^kf}{dt^k}\bigg|_{t=0}$$
Now suppose $f(t)=\sin(t)$. Then the derivatives cycle through 0,1,-1,0, and so on.
The sum doesn't converge, yet the integral is 1/2.
What's going on? Is there a function other than a finite sum of polynomials for which this works?