# pointwise and uniform convergence problem

• Mar 12th 2011, 07:18 PM
wopashui
pointwise and uniform convergence problem
Determine the convergence, both pointwise and uniform, on [0,1] for each of the following sequences:

1) $\displaystyle S_n(x)= n^2x^2(1-cos1/[nx]), x not= 0; S_n(0)=0$.

2) $\displaystyle S_n(x)= nx/(x+n)$.

3) $\displaystyle S_n(x)= (1-cosnx)/nx, x not= 0; S_n(0)= 0$.

4) $\displaystyle S_n(x)= nsin(x/n)$.

I'm really struglling with these uniformly convergence problems, any help would be appreciated
• Mar 12th 2011, 11:21 PM
FernandoRevilla
For example, did you have any problems finding $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ ?
• Mar 13th 2011, 11:17 AM
wopashui
Quote:

Originally Posted by FernandoRevilla
For example, did you have any problems finding $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ ?

$\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ actually, i'm not sure what is the difference between $\displaystyle S(x) and S_n(x)$
• Mar 13th 2011, 09:42 PM
FernandoRevilla
Quote:

Originally Posted by wopashui
$\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ actually, i'm not sure what is the difference between $\displaystyle S(x) and S_n(x)$

For example 2):

$\displaystyle S(x)=\displaystyle\lim_{n\to +\infty}\dfrac{x}{(x/n)+1}=\dfrac{x}{0+1}=x$

So, $\displaystyle S_n:[0,1]\to \mathbb{R}$ converges pointwise to $\displaystyle S:[0,1]\to \mathbb{R},\; S(x)=x$. Try to find $\displaystyle S(x)$ for the rest.