A few months ago I learned that Euler's gamma function: $$\int_0^{\infty} e^{-t}t^{x-1}dt$$ converges when $x \gt 0$. Then few weeks ago I find out from theory on power series that convergence for harmonic series $\sum {1 \over n^\alpha}$ can be found for $\alpha \in \mathbb{R}$ by using a integral criteria which says that series converges if and only if improper integral $\int_{1}^\infty {dx \over x^\alpha}$ converges, and that is right for $\alpha \gt 1$.
What I am trying to do here, is to implement integral criteria from power series to gamma function, but for ${1 \over t^{1 - x} }$ it doesn't seem to work since for $1-x \gt 1$ or $x \lt 0$, gamma function diverges, and i expect from integral criteria to converge. What am I doing wrong?
Let $u=t^{x/2}$ and assume $x>0$,
$$\int_1^\infty t^{x-1}e^{-t}\ dt=\frac2x\int_1^\infty e^{-u^{2/x}}\ du$$
As $x>0$, $2/x>0$, and so it is easy enough to check that $\int_1^\infty e^{-u^{2/x}}\ du$ will converge.
For $x<0$,
$$\int_1^\infty t^{x-1}e^{-t}\ dt=\frac2x\int_1^0e^{-u^{2/x}}\ du$$
And clearly this is finite.
For $x=0$, convergence is also trivial.
But for $\int_0^1t^{x-1}e^{-t}\ dt$, notice that if $x\ge1$, the integral is obviously finite.
For $1>x>0$, see that
$$\int_0^1t^{x-1}e^{-t}\ dt<\int_0^1t^{x-1}\ dt$$
And so this converges trivially. But for $x\le0$,
$$\int_0^1t^{x-1}e^{-t}\ dt>\int_0^1t^{x-1}e^{-1}\ dt$$
And so this diverges trivially.