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What is the result of this factorial sum?
I need to know the result of the following sum:
sum[ x^n / (k-n)! ] for n=1 to infinity,
k is a constant positive integer, k>n
x is constant, 0<x<1
I know that sum[ x^n / n! ] for n= 0 to infinity is e^x, still I cannot get the result of the sum mentioned above. Any ideas?
Thank you
1 Answer
- 1 decade agoFavorite Answer
In which standard are you?
do you know about Gamma functions.If yes then you may be able to solve it and i think if k is a constant integer then at certain point k-n will become negative and hence (k-n)!=1 for all n>=k
Do try to solve it yourself
however sum[x^n/(n-k)!] for n=1 to infinity is
e^x* x^k - (e^x *x^k *Gamma[1 - k, x])/Gamma[1 - k]
