uniform convergence exercise

[CarlosP]

I have had some problems solving unfirome convergence exercises, and I am not able to solve this. Can anybody help me?
I will translate to English:
Study the uniform convergence of the series on set E

 

[pka]

You need to show what you have tried. This site is not for doing homework.
Rather it is a help site. We can only help if you show effort.
Have you explored using the root test?

 

[CarlosP]

You need to show what you have tried. This site is not for doing homework.
Rather it is a help site. We can only help if you show effort.
Have you explored using the root test?

I can solve the part that show that converge. But don't know the next stop to show the uniform convergence:
Let z be different from 0,then,
[MATH] -1 <= (z^n)/(n* log^2(n+1)) <= (z^n)/n. [/MATH]Since [MATH](z^n)/n[/MATH] converge ( by using the D'Alembert Test [MATH] lim_{n->inf} u_{(n+1)} / u_n = l < 1[/MATH] ) then the series converge on set E .

 

[pka]

I can solve the part that show that converge. But don't know the next stop to show the uniform convergence:
Let z be different from 0,then, [MATH]-1 <= (z^n)/(n* log^2(n+1)) <= (z^n)/n.[/MATH]Since [MATH](z^n)/n[/MATH] converge ( by using the D'Alembert Test [MATH] lim_{n->inf} u_{(n+1)} / u_n = l < 1[/MATH] ) then the series converge on set E .

Well I guess you are waiting for the answer? That is not the way we work here.
Can you show that for \(\displaystyle \forall z\in E\) then \(\displaystyle \left(f_n(z)\right)\) forms a Cauchy sequence?