# Thread: Uniform convergence of a sequence of functions

1. ## 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?

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?

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

Can the following sequence of function be differentiated?

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 $\displaystyle \sum_{n=1}^\infty\frac{1}{x^2+n^2}$. This one is easy to prove (cf. normal convergence).

3. I think I got it, thanks.

BTW, isn't the derivative $\frac{1}{\frac{{x}^{2}}{{n}^{2}}+{n}^{2}}$ ? (It doesn't make a difference anyway )

4. Originally Posted by BMWM5
I think I got it, thanks.

BTW, isn't the derivative $\frac{1}{\frac{{x}^{2}}{{n}^{2}}+{n}^{2}}$ ? (It doesn't make a difference anyway )
Yes, sorry. And indeed, the proof procedes the same way.