Convergence of the following series as N goes to infty

[Marin]

hi there!

I want to show the convergence of the following series as N goes to infty.

\displaystyle{\sum_{k=0}^N}\frac{x^k}{k!}-\frac{n!x_n^k}{k!(n-k)!n^k},

x_n is a sequence such that. lim(n->oo)x_n = x, but I´m interested in big N

I ´m not allowed to use the limit definition of exp(x)

What I want to do (but am not sure if it´s correct) is to separate the sum before taking the limit N->oo and after that take it, so that the first term converges to exp(x) an the convergence of the second term I can show with the ratio test

 

[lurflurf]

Are you sure N and n are not related?
What is your convention when k>N?
The usual convention is that nCk=0

 

[JG89]

I don't know if this will help, but did you notice that the \frac{n!}{k!(n-k)!} is the binomial coefficient? Might be worth something...

 

[Marin]

lurflurf, sorry I forgot to say that n>=N, but it really comes up to N not n.

JG89, thanks, it is also given as the binomial coefficient, I wrote it that way because I don´t know how to write it in latex language :(


could you say if I can do it the way I described it, or it´s somehow wrong to do like that?