Gamma Infinite Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)}{n!}=0$ Avoiding the analytic continuation of extended binomial theorem, 
$$ \sum_{n=0}^{\infty}\frac{\Gamma(n+z)}{n!}\,x^n = \frac{\Gamma(z)}{(1-x)^z} \quad\colon\space |x|\lt1 $$ 

How to prove: 
  $$ \sum_{n=0}^{\infty}\frac{\Gamma(n+s)}{n!} = 0 \quad\Rightarrow\, \frac{s}{1!}+\frac{s(s+1)}{2!}+\cdots = -1 \quad\colon\space Re\{s\}\lt0 $$

 A: We find that
$$ \sum_{k=0}^{n} \frac{\Gamma(k+s)}{k!} = \frac{\Gamma(n+1+s)}{n!s} \tag{*}$$
for all $n = 0, 1, 2, \cdots$. Indeed, this is easily proved by the mathematical induction:


*

*When $n = 0$, it boils down to the equality $\Gamma(s) = \frac{\Gamma(1+s)}{s}$, which is of course true.

*Assuming that $\text{(*)}$ is true for $n \geq 0$, then
\begin{align*}
\sum_{k=0}^{n+1} \frac{\Gamma(k+s)}{k!}
&= \frac{\Gamma(n+1+s)}{n!s} + \frac{\Gamma(n+1+s)}{(n+1)!} \\
&= \frac{(n+1+s)\Gamma(n+1+s)}{(n+1)!s}
 = \frac{\Gamma(n+2+s)}{(n+1)!s}
\end{align*}
Therefore $\text{(*)}$ is true for all $n \geq 0$. Now the conclusion follows by taking $n\to\infty$. (Stirling's formula is enough for this purpose.)

Remark. The identity $\text{(*)}$ becomes more natural once we recognize it as a disguise of the famous formula
$$ \sum_{k=0}^{n} \binom{k+s-1}{k} = \binom{n+s}{n}. $$
When $s$ is a positive integer, this indeed follows from the hockey-stick argument.
A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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With $\ds{\Re\pars{s} < 0}$:

\begin{align}
\left.\sum_{n = 0}^{\infty}{\Gamma\pars{n + s} \over n!}
\,\right\vert_{\ \Re\pars{s}\ <\ 0} & =
\pars{s - 1}!\sum_{n = 0}^{\infty}{n + s - 1 \choose n} =
\pars{s - 1}!\sum_{n = 0}^{\infty}{-s \choose n}\pars{-1}^{n}
\\[5mm] & =
\pars{s - 1}!\,\bracks{1 + \pars{-1}}^{\,-s} = \bbx{\ds{0}}
\end{align}
A: \begin{eqnarray*}
\Gamma(z) = \int_0 ^{\infty} x^{z-1} e^{-x} dx
\end{eqnarray*}
So
\begin{eqnarray*}
\sum_{n=0}^{\infty} \frac{ \Gamma(n+s)}{n!} =\sum_{n=0}^{\infty} \int_0 ^{\infty} \frac{x^{n+s-1} e^{-x}}{n!} dx
\end{eqnarray*}
Now invert the sum & integral
\begin{eqnarray*}
\int_0 ^{\infty} x^{s-1} \sum_{n=0}^{\infty}  \frac{x^{n} }{n!}  e^{-x}dx =\int_0 ^{\infty} x^{s-1} dx = \left[ \frac{x^s}{s} \right]_0 ^{\infty}
\end{eqnarray*}
Now $x^s$ will tend to zero provided $ Re(s)< 0$.
\begin{eqnarray*}
\Gamma(s+1) = s \Gamma(s) \\
\Gamma(s+2) = s(s+1) \Gamma(s) \\
\Gamma(s+n) = s(s+1) \cdots (s+n-1)  \Gamma(s)
\end{eqnarray*}
Divide the equation by $\Gamma(s)$ and move the first term to the right hand side.
