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

Amaelle
Messages
309
Reaction score
54
Homework Statement
Loook at the image
Relevant Equations
Power series.
Raduis of convergence.
Greetings
I have some problems finding the correct result
1629716024934.png

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:
1629716182475.png

1629716392658.png

  • 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
1629716620401.png

1629716656750.png


I would like to know where is my mistake
thank you!
 
Physics news on Phys.org
I am not sure about your notation \log^{2021}x. Is it
\log_{2021}x or
(\log x)^{2021}?
 
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 ?
 
  • Like
Likes Delta2 and Amaelle
anuttarasammyak said:
I am not sure about your notation \log^{2021}x. Is it
\log_{2021}x or
(\log x)^{2021}?
(\log x)^{2021}?
 
anuttarasammyak said:
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!
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top