Determine Series Convergence Given Convergence of a Power Series

[BraedenP]

Homework Statement

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

$$\sum_{n=0}^{\infty}c_nx^n$$ 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:
$$\sum_{n=0}^{\infty}c_n$$

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 $c_n$ 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!

 

[HallsofIvy]

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.

 

[LCKurtz]

And to add to Halls' hint, remember a power series converges absolutely on the interior of the interval of convergence.

 

[BraedenP]

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?

 

[LCKurtz]

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.