How to simplify this summation to an incomplete gamma function

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PureStorm
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Could someone please explain why the following sum simplifies to the following?

Mathematical formula: 2 + sum from k = 2 to n of (n − 1)! / ((n − k)! × n^(k − 1)).


=
Mathematical expression e^n n^(-n) Γ(n+1, n), with the incomplete gamma function written as Γ(n+1, n).


As far as I can see, this sum does not correlate to the formula for incomplete gamma function as a sum. I'd appreciate any help as the incomplete gamma function is somewhat beyond the scope of my current course.



P.S. I managed to do it eventually. The key was recognising that n-k could be substituted for k as k varies from 0 to n.
 
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