Welcome to our community

Be a part of something great, join today!

Uniform convergence

Jan 31, 2012
54
Hello!


A little problem:


With the given series,


$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,


why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?


Ok, I know that $Y(x)$ is u.c by M-test:

$$\max{|x\ln{x}|}=\frac{1}{e}$$

And,

$$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$

Is converges! But why only in $(0,1]$ ?



Thank you!
 

girdav

Member
Feb 1, 2012
96
We need some clarifications. First, we talk about uniform convergence on a set, not only at one point (in this case, it's pointwise convergence). Where do you take the $\max$?
 
Jan 31, 2012
54
Last edited:

girdav

Member
Feb 1, 2012
96
In this case, the $\max$ is infinite. But if you take it on $(0,1]$, you will notice that the series is normally convergent on $(0,1]$.
 
Jan 31, 2012
54
In this case, the $\max$ is infinite. But if you take it on $(0,1]$, you will notice that the series is normally convergent on $(0,1]$.
Normally convergence is not enough for this problem that I have, I need integrate that sum,- I must first prove the U.C. on (0,1].

I'll rephrase my question: is that the only interval,(0,1], that Y(x) U.N in ?
 

girdav

Member
Feb 1, 2012
96
You have the normal (hence uniform) convergence on each interval of the form $(0,a]$, $a>0$.