Solving the Radius of Convergence of a Periodic Power Series

[bcucinel]

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.

 

[ideasrule]

What if x is negative? Then you'd have an alternating series. You have to consider where this series converges.

 

[Dick]

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.