Convergence of the following series as N goes to infty

In summary, the conversation is about showing the convergence of a series involving x_n and N as N goes to infinity. However, the limit definition of exp(x) cannot be used. The proposed method is to separate the sum before taking the limit and then showing the convergence of the second term using the ratio test. The relationship between N and n is clarified and the use of binomial coefficients is also mentioned. There is a question about whether the proposed method is correct or not.
  • #1
Marin
193
0
hi there!

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

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

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
 
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  • #2


Are you sure N and n are not related?
What is your convention when k>N?
The usual convention is that nCk=0
 
  • #3


I don't know if this will help, but did you notice that the [tex]\frac{n!}{k!(n-k)!}[/tex] is the binomial coefficient? Might be worth something...
 
  • #4


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?
 

FAQ: Convergence of the following series as N goes to infty

What is the concept of "convergence" in a series?

The concept of convergence in a series refers to whether or not the terms in the series approach a finite limit as the number of terms approaches infinity. In other words, it determines whether the sum of the infinite terms in a series will result in a finite value or not.

What is the significance of the limit as N approaches infinity in the convergence of a series?

The limit as N approaches infinity is the key factor in determining the convergence of a series. If the limit exists and is a finite value, then the series is said to converge. If the limit does not exist or is infinite, the series diverges.

How is the convergence of a series determined?

The convergence of a series can be determined by using various tests such as the ratio test, root test, comparison test, and integral test. These tests analyze the behavior of the terms in the series and check if they approach a finite limit as N goes to infinity.

What does it mean if a series is divergent?

A series is divergent if the limit of its terms does not approach a finite value as the number of terms approaches infinity. This means that the sum of the infinite terms in the series will not result in a finite value and the series is said to be divergent.

Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. The concept of convergence and divergence are mutually exclusive. A series can either converge or diverge, but not both at the same time. It is important to note that a series may not converge or diverge at all, in which case it is considered to be oscillating or undetermined.

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