Given Cn4^n is convergent as n goes from 0 to infinity:

(i) how is Cn(-2)^n as n goes from 0 to infinity convergent?

(ii) how is Cn(-4)^n as n goes from 0 to infinity not convergent?

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- Oct 5th 2010, 04:49 PMTaurus3comparing series
Given Cn4^n is convergent as n goes from 0 to infinity:

(i) how is Cn(-2)^n as n goes from 0 to infinity convergent?

(ii) how is Cn(-4)^n as n goes from 0 to infinity not convergent? - Oct 5th 2010, 09:07 PMtonio
- Oct 5th 2010, 10:08 PMTaurus3
cn is just a constant, or a coefficient of the series.

- Oct 6th 2010, 09:13 AMKrizalid
even being a constant doesn't converge.

- Oct 6th 2010, 05:38 PMtonio
- Oct 7th 2010, 06:27 AMHallsofIvy
The power series always has a "radius of convergence", R, and converges inside that radius- on the interval [-R, R] while diverging outside that interval. Since converges, 4 is either inside that interval or is an endpoint (R= 4).

In either case, since -4< -2< 4, -2 is inside the interval of convergence and converges.

I had first thought that "if converges, then the series must converge but then realized that I was assuming that is**positive**- and that is not given. If the radius of convergence is 4, knowing that the power series converges at one endpoint of the interval of convergence does not tell us whether the series converges at the other endpoint.

If we know only that converges, then we know that converges but do not know whether converges or not.

If we**also**know that for all n,**then**we know that converges (absolutely).