# Uniform convergence - Series of functions

1. Jun 16, 2010

### estro

1. The problem statement, all variables and given/known data
$$\mbox{Check whether } \sum_{n=0}^\infty \frac {1}{e^{|x-n|}} \mbox{ is uniform convergent where its normaly convergent}$$

3. The attempt at a solution

$$\mbox{I choose } \epsilon = 1/2$$

$$a_n=\frac {1}{e^{|x-n|}}\ ,\ b_n= \frac {1}{n^2}$$

$$\lim_{n\rightarrow\infty} \frac {a_n}{b_n}=0\ \Rightarrow\ \sum_{n=0}^\infty a_n \mbox{ is convergent for all x \in R.}$$

$$f_k(x)=\sum_{n=0}^{k} a_n\ ,\ f(x)=\lim_{k\rightarrow\infty} \sum_{n=0}^{k} a_n$$

$$x_n=n\Rightarrow\\sup_{x_n \in R} |f_k(x_n)-f(x_n)|\geq\sum_{n=k+1}^\infty \frac {1} {e^{|n-n|}}=1>\epsilon\ \Rightarrow \mbox{ is not uniformly convergent \inR}$$

What do you think?
[I left out technical details to present my idea more clearly.]

Last edited: Jun 16, 2010