Understanding Parenthesis Style in Power Series Questions

[melissax]

Hello,
My question is about power series.
In most of questions i can find points with ratio test.
But when i check points i can't understand style of parenthesis.
Is there easy way?
For example:$\sum(n^(3)*(x-5)^n)$
I found check points 4<x<6
How can i decide to the parenthesis will be closed or open?

Thank you

 

[Millennial]

I don't understand anything from this. What paranthesis? What style? And how do you find points with the ratio test? It is supposed to give you a boundary condition on x, you aren't supposed to obtain a point.

 

[melissax]

I uploaded question this is not a homework i just trying to understand how we decide to the
boundaries of covergence. Also i don't know how i can write mathematicaly here.

Lim n->inf |((n+1)^(3)*(x-5))^(n+1)/(n^(3)*(x-5)^n)

Lim n->inf |(x-5)*(((n+1)/n)^3)| after x-5 this part equal to the 1

-1<|x-5|<1
4<x<6

Then i tried to check points;
I put this points to the question
Ʃ[0,inf]n^(3)*(x-5)^n = Ʃ[0,inf]n^(3)*(4-5)^n
Ʃ[0,inf]n^(3)*(-1)^n
After this part how i will decide parantesis will be closed or open?

 

[mfb]

Also i don't know how i can write mathematicaly here.

If you know LaTeX: [tex]a^2+b^2[/tex] gives $$a^2+b^2$$

After this part how i will decide parantesis will be closed or open?

You mean the interval borders? You have to check them separately. Just set x to the corresponding value, and see if the series converges.