Solving the Radius of Convergence of a Periodic Power Series

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SUMMARY

The radius of convergence for the periodic power series Σanxn, where an=1,2,3,1,2,3,... with a period of p=3, is determined to be r=1. The series converges when the absolute value of x is less than 1, leading to the interval of convergence being (-1,1). It is crucial to recognize that while the series has geometric characteristics, it is not purely geometric, necessitating further analysis for convergence, especially when considering negative values of x.

PREREQUISITES
  • Understanding of power series and their convergence criteria
  • Familiarity with periodic sequences and their properties
  • Knowledge of geometric series and their convergence
  • Basic calculus concepts related to limits and absolute values
NEXT STEPS
  • Study the properties of periodic power series in depth
  • Learn about the Ratio Test for determining convergence
  • Explore the concept of alternating series and their convergence criteria
  • Investigate the implications of different coefficient patterns on convergence
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Mathematics students, educators, and anyone studying series convergence, particularly those focused on power series and periodic functions.

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Homework Statement



Consider the power series

Σanxn = 1+2x+3x2+x3+2x4+3x5+x6+…

in which the coefficients an=1,2,3,1,2,3,1,... are periodic of period p=3. Find the radius of convergence.

Homework Equations





The Attempt at a Solution


My attempt at a solution was to first state that the series was geometric as r=x
In order for this series to converge, the absolute value of r must be less than 1.
|r|<1
Therefore |x|<1.
Based on this, the interval of convergence is (-1,1) and the radius is r=1.

Based on my solution, I did not take into account an... If there is another solution I would need to take an into account to solve the radius of convergence please let me know.
 
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What if x is negative? Then you'd have an alternating series. You have to consider where this series converges.
 
But the series isn't a geometric series. So you don't know |x|<1 yet. You can split it up into some series that are geometric, though.
 

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