# Uniform convergence of a sequence of functions

• Apr 17th 2010, 12:15 PM
BMWM5
Uniform convergence of a sequence of functions
Hi, I was hoping someone might be able to help me out.

Can the following sequence of function be differentiated?

http://www.sitmo.com/gg/latex/latex2...%7Bn%5E2%7D%29

As far as I understood, I have to check continuity and uniform convergence. I think it fails to satisfy the latter. How can I explain it?
• Apr 17th 2010, 12:29 PM
Laurent
Quote:

Originally Posted by BMWM5
Hi, I was hoping someone might be able to help me out.

Can the following sequence of function be differentiated?

http://www.sitmo.com/gg/latex/latex2...%7Bn%5E2%7D%29

As far as I understood, I have to check continuity and uniform convergence. I think it fails to satisfy the latter. How can I explain it?

Actually, you need uniform convergence of the term-by-term derivative $\sum_{n=1}^\infty\frac{1}{x^2+n^2}$. This one is easy to prove (cf. normal convergence).
• Apr 17th 2010, 12:46 PM
BMWM5
I think I got it, thanks.

BTW, isn't the derivative http://latex.codecogs.com/gif.latex?...5E%7B2%7D%7D$$? (It doesn't make a difference anyway (Smirk)) • Apr 17th 2010, 12:49 PM Laurent Quote: Originally Posted by BMWM5 I think I got it, thanks. BTW, isn't the derivative http://latex.codecogs.com/gif.latex?...5E%7B2%7D%7D$$ ? (It doesn't make a difference anyway (Smirk))

Yes, sorry. And indeed, the proof procedes the same way.