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Determine Series Convergence Given Convergence of a Power Series

  • Thread starter BraedenP
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Homework Statement



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]

Homework Equations





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!
 

Answers and Replies

  • #2
HallsofIvy
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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.
 
  • #3
LCKurtz
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And to add to Halls' hint, remember a power series converges absolutely on the interior of the interval of convergence.
 
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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?
 
  • #5
LCKurtz
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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?
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.
 

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