Power series problem

[Bliman]

Hi all,
I have this power series problem and the convergence of it that I don't understand is the following one the summation of
(k(k+1)(k+2)***(k+n-1)x^n)/n! from n=0 to infinity
Now you solve this with the ratio test and you do this with lim as n approaches infinity of the absolute value of (an+1)/an and solve it for less than one.
Now what I don't get is why you don't do (k+(n+1)-1) but instead skip over it and do this (k+n-1)(k+n). Why not do the (n+1) directly instead of leaving (k+n-1) alone and multiplying it then with (k+n)?
I hope it is clear because I don't find how you put mathematical symbols in the question (the read before posting didn't really help).
Also these *** always confuses me, sometimes you have to put them in factorials but that is super confusing for me. Where can I learn more how to handle this?
Thanks for helping me
Sorry but I posted this again. Because my other thread wasn't visible. And now that I post this it is again visible. Weird. Also when I looked at how many messages I posted it gave me 0. So this one can be deleted. I really don't know why this happened.

 

[Steven G]

Why keep k+n-2 and k+n-3 all the way down to k+3, k+2, k+1 and k?? Because they are there! What is the number before k+n?

Do you realize that k(k+1)(k+2)...(k+n) = k(k+1)(k+2)...(k+n-1)(k+n+1). They BOTH contain all integers from k all the way to k+n. The number before k+n just happens to be k+n-1.