Sum of Gamma function

Question

$$ \sum_{t=1}^{n}{\frac{\Gamma(t)}{\Gamma(a+t)}} $$ and the result of wolframalpha is result

and I don't know the detail process about it

Answer 1

Notice that \begin{gather*} \frac{\Gamma ( r)}{\Gamma ( a+r)} =\frac{1}{a-1} \cdotp \frac{( a-1) \Gamma ( r)}{\Gamma ( a+r)}\\ =\frac{1}{a-1} \cdotp \frac{( a-1) \cdotp r!}{( a+r-1) !} \end{gather*} Again, \begin{gather*} \frac{r!}{( a+r-1) !} -\frac{( r+1) !}{( a+r) !} =\frac{r!}{( a+r-1) !}\left( 1-\frac{r+1}{a+r}\right) =\frac{r!}{( a+r-1) !} \cdotp \frac{a-1}{( a+r)}\\ =\frac{r!\cdotp ( a-1)}{( a+r) !} \end{gather*} Can you now complete the prof using the above?