Finding the Radius of Convergence for Y=6x+16 - Troubleshooting and Solution

[Amaelle]

Homework Statement

Loook at the image

Relevant Equations

Power series.
Raduis of convergence.

Greetings
I have some problems finding the correct result

My solution:
I puted Y=6x+16
so now will try to find the raduis of convergence of Y
so let's calculate the raduis criteria of convergence:

• We know that Y=6x+16
• Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not the solution!

Here is the solution of the book

I would like to know where is my mistake
thank you!

 

[anuttarasammyak]

I am not sure about your notation $$\log^{2021}x$$. Is it
$$\log_{2021}x$$ or
$$(\log x)^{2021}$$?

 

[anuttarasammyak]

5 for 6x+16 and 5/2 for 3x+8 seem not different. What is the rule to define t for t^n series for convergence ?

 

[Amaelle]

I am not sure about your notation $$\log^{2021}x$$. Is it
$$\log_{2021}x$$ or
$$(\log x)^{2021}$$?

$$(\log x)^{2021}$$?

 

[Amaelle]

5 for 6x+16 and 5/2 for 3x+8 seem not different. What is the rule to define t for t^n series for convergence ?

I realize that I have the same confusion, I thought we were looking for the domain where x is convergent?
I understand I got the same results implicitely but did the book go for that particular set up?
thanks a million!