Radii of convergence for exponential functions


by Bionerd
Tags: convergence, exponential, functions, radii
Bionerd
Bionerd is offline
#1
Dec9-07, 09:03 AM
P: 12
1. The problem statement, all variables and given/known data

I have to find the radius of convergence for each: (A) e^3x (B) xe^(-x)^2 (C) (e^x-1)/x


2. Relevant equations

So I used the ratio test for each. I'll only write out the answer I got since I'm pretty sure I did them correctly.

(A) abs. value (x) lim n--> infinity 3/(n+1)

(B) abs. value (x^2) lim n--> infinity 1/(n+1)

(C) abs. value (x) lim n--> infinity 1/(n+1)

3. The attempt at a solution

My question isn't so much about how to do them (I can always go back and redo the ratio test if my answers are wrong) but about what to claim as the radius of convergence. For all of these, it seems to be infinity, and I know e^x has a radius of convergence of infinity. Do all of these really have the same radius of convergence?
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CompuChip
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#2
Dec9-07, 09:23 AM
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I don't feel like doing the calculations either, but your answers are correct. In fact, I think the series of anything of the form (polynomial)x(exponential) will always converge everywhere (you can try proving it, probably it's easy as it just shifts the powers in the series expansion up or down, so it shifts the coefficients a_n, but that doesn't matter for thhe limit n -> infinity). Actually the only case which needs special attention is then the third one, but you see quickly enough that at the dangerous point (x = 0) nothing special happens (using L'Hopital, the limit x -> 0 exists, etc) and indeed your calculation shows that despite appearances there is no problem there either.
HallsofIvy
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#3
Dec9-07, 01:08 PM
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First, look at the way you have posed the question. In general, functions don't have a "radius of convergence"! Power series have a radius of convergence. So you are really asking "what is the radius of convergence of a power series converging to these functions?" But the power series has to be about a specific "central point": (x-a)n for some a. If a power series does not converge for all x (i.e. its radius of convergence is not infinity), then the radius of convergence will depend upon the central point- which you are not given here! In general, a power series will converge "as long as there is no reason not to" which basically means "as long as the function is infinitely differentiable. As CompuChip says, it is obvious, for all except possibly the last, that, since ex is infinitely differentiable, so are these. And, again as CompuChip says, it is clear, by being careful with limits at x= 0, that the last is also infinitely differentiable.


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