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

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SUMMARY

The discussion focuses on finding the radius of convergence for the function Y=6x+16. The user initially calculated the radius as 5/3 based on the interval -21/6 <= x <= -11/6, but this was incorrect. The confusion arose from the notation used for logarithmic functions and the rules for defining the variable t in t^n series for convergence. Clarification on these points is essential for accurately determining the radius of convergence.

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Amaelle
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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!
 
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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 ?
 
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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!
 

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