# uniform convergence

• Nov 4th 2009, 02:13 PM
xyz
uniform convergence
http://i291.photobucket.com/albums/l...Capture123.jpg

is this true or false? Explain why. I think it is not true because the interval is unbounded. However, I cannot explain why.
• Nov 4th 2009, 05:32 PM
TheEmptySet
Quote:

Originally Posted by xyz
http://i291.photobucket.com/albums/l...Capture123.jpg

is this true or false? Explain why. I think it is not true because the interval is unbounded. However, I cannot explain why.

It is true. Here is a hint

$\displaystyle f_n(t)=\frac{1}{t\ln(n)}$ the largest value of any sequence member is given when $\displaystyle t=1$ i.e $\displaystyle f_n(1)\gef_n(t)$ for all $\displaystyle t \in [1,\infty]$
• Nov 4th 2009, 06:05 PM
xyz
uniform convergence
If thats true, I am not quite sure how this problem is done. The complete problem is

http://i291.photobucket.com/albums/l.../Capture34.jpg

I appreciate your help a lot. The next thing, is to take limit of n on both sides. Does that mean right hand limit at 0 and left hand limit at 0? Please let me know how the problem ends up with 1 as value of integral when it should be 0?
• Nov 4th 2009, 06:32 PM
redsoxfan325
It's true that uniform convergence allows you to interchange the limit and the integral, but only on a bounded interval $\displaystyle [a,b]$. (The proof of the theorem uses the value $\displaystyle b-a$, which makes no sense if $\displaystyle b=\infty$.)