Radius of Convergence Problem with Solution Attempt

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SUMMARY

The discussion focuses on solving the Radius of Convergence problem, specifically addressing the convergence of the series related to the term involving factorials. The user initially struggles with the term n! in the numerator but receives guidance to check the (n+1)! term, which should be squared. Ultimately, the conclusion reached is that the radius of convergence R is -1, leading to the open interval of convergence defined as 8 < x < 6.

PREREQUISITES
  • Understanding of series convergence and divergence
  • Familiarity with factorial notation and its properties
  • Knowledge of the Ratio Test for determining convergence
  • Basic algebraic manipulation skills
NEXT STEPS
  • Review the Ratio Test for series convergence
  • Study the properties of factorials in mathematical series
  • Explore the concept of open intervals in real analysis
  • Practice additional problems on Radius of Convergence
USEFUL FOR

Students studying calculus, particularly those focusing on series and convergence, as well as educators looking for examples of Radius of Convergence problems.

jegues
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Homework Statement



See figure attached for problem statement as well as my attempt.

Homework Equations





The Attempt at a Solution



See 2nd figure attached.

I don't know how to rid myself of that last n! I've got kicking around in the numerator.

Any ideas?
 

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Check your an+1 term. The [(n+1)!] should be squared.
 
jav said:
Check your an+1 term. The [(n+1)!] should be squared.

Doh!

Thank you!
 
Just to make sure I finished the question off right, we should find that R = -1, correct?

Then the open interval of convergence is,

|x-7| &lt; -1

or 1 &lt; x-7 &lt; -1 or 8 &lt; x &lt; 6
 

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