babyrudin
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Hello all! In solving some math problems, I encountered the following sum:
[tex]\sum_{k=1}^{r+1} kb \frac{r!}{(r-k+1)!} \frac{(b+r-k)!}{(b+r)!}. \quad \mbox{(eqn.1)}[/tex]
Now, I have asked Maple to calculate the above sum for me, and the answer takes a very simple form:
[tex]\frac{b+r+1}{b+1}. \quad \mbox{(eqn.2)}[/tex]
My question is, does anyone know how to go from (eqn.1) to (eqn.2)? I am really bad at working with factorials, and so far I am not getting close. Maybe there are some results and properties of factorials and summation that can be used to simplify the above?
[tex]\sum_{k=1}^{r+1} kb \frac{r!}{(r-k+1)!} \frac{(b+r-k)!}{(b+r)!}. \quad \mbox{(eqn.1)}[/tex]
Now, I have asked Maple to calculate the above sum for me, and the answer takes a very simple form:
[tex]\frac{b+r+1}{b+1}. \quad \mbox{(eqn.2)}[/tex]
My question is, does anyone know how to go from (eqn.1) to (eqn.2)? I am really bad at working with factorials, and so far I am not getting close. Maybe there are some results and properties of factorials and summation that can be used to simplify the above?