Uniform convergence vs pointwise convergence

Hi guys,

Can someone please explain to me the advantages of uniform convergence over pointwise convergence? I was told that pointwise convergence does not preserve continuity while uniform convergence does. Also, pointwise convergence depends on x while uniform convergence doesn't. But whats the big deal about it? I am not getting the whole picture of it. We are learning Fourier Series by the way.

Thanks.

Re: Uniform convergence vs pointwise convergence

can somebody help me with this problem please??

Show that the sequence {f_n} where f_n=nxe^(-nx^2 ) for n=1,2,3,… converges pointwise in [0,1].Is the convergence uniform?Justify

Re: Uniform convergence vs pointwise convergence

Quote:

Originally Posted by

**chath** Show that the sequence {f_n} where f_n=nxe^(-nx^2 ) for n=1,2,3,… converges pointwise in [0,1].Is the convergence uniform?Justify

Prove that $\displaystyle f(x)=\lim_{n\to +\infty}f_n(x)=0$ for all $\displaystyle x\in [0,1]$ and $\displaystyle \int_0^1 f(x)\;dx\neq \lim_{n\to +\infty}\int_0^1f_n(x)\;dx$.

P.S. It is better a new thread for every new problem.