1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Determine Series Convergence Given Convergence of a Power Series

  1. Oct 10, 2011 #1
    1. The problem statement, all variables and given/known data

    I am asked to comment on the convergence/divergence of three series based on some given information about a power series:

    [tex]\sum_{n=0}^{\infty}c_nx^n[/tex] converges at x=-4 and diverges x=6.

    I won't ask for help on all of the series, so here's the first one:
    [tex]\sum_{n=0}^{\infty}c_n[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I tried reasoning that the question is suggesting a convergence interval of (-14,6) for the power series (I took -4 as the center, and 6 as the right-hand side) but the more I read the question, I don't think that's what it's suggesting. It's just commenting about divergence at those two exact points.

    So now I'm stuck. Am I supposed to figure out the value of [itex]c_n[/itex] and work out the divergence of other series that way, or is there some way for me to compare these series using what I know about their centers of convergence (both are centered around 0).

    Guidance would be awesome!
     
  2. jcsd
  3. Oct 10, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    A power series always has an 'interval of convergence'. Since the given power series converges at -4, that interval of convergence must be at least form -4 to 4. And x= 1 is inside that interval.
     
  4. Oct 10, 2011 #3

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    And to add to Halls' hint, remember a power series converges absolutely on the interior of the interval of convergence.
     
  5. Oct 10, 2011 #4
    Okay, thanks guys :) That makes perfect sense to me if I accept the fact that convergence is guaranteed along (-4,4). However I don't think I quite understand why convergence is guaranteed along that interval.

    Is it as simple as saying that since it's centered around 0 and converges at x=-4, then the radius of convergence is 4, and thus is must also converge at x=4, forming the interval of convergence?
     
  6. Oct 10, 2011 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Almost. But the radius of convergence might be greater than 4 so you don't know it is exactly 4. All you know is the radius of convergence isn't less than 4 or greater than 6. And even if the radius of convergence was 4, you wouldn't know it converged at 4.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Determine Series Convergence Given Convergence of a Power Series
Loading...