Incomplete gamma function question

[rman144]

I need to show whether or not:

|\gamma(s,1)| converges for 0<Re(s)<1.

Does anyone know of an expansion for this that would prove convergence?

[HallsofIvy]

What do you mean by "converges"? There is no limit here.

[g_edgar]

The improper integral
\Gamma \left( a,z \right) =\int _{z}^{\infty }\!{{\rm e}^{-t}}{t}^{a-
1}{dt}
converges for Re(a) > 0 by comparison with
\int _{z}^{\infty }{t}^{a-1}{dt}.
Is that the question?