Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general.


I considered the function

$ f(x) = - \frac {2x^2+3x}{(x+1)^2} $

I expressed this by a powerseries $ f_1(x) = -3x + 4x^2 - 5x^3 + 6x^4 - \ldots $ and stated without the derivation that this is also $ f_2(x) = \frac {-2}{1} -\frac {-1}{x} + 0 - \frac {1} {x^3} + \frac {2}{x^4} - \ldots + \ldots $
and - well: hell, - don't see it now how I did it.

What was interesting to me was, that after looking for the fixpoints $ x_0=0, x_{1,2} =-2 $ the range of convergence in the expression by $f_1$ is obviously $ |x|<1 $ limited to the unit-interval but in that by $f_2$ it is infinity and $ |x|>1 $ .

I would like to be able to translate also other powerseries into an $f_2$-type-expression. (I remember to have read a remark of "expanding a powerseries at infinity" but have never seen an explanation of this - so this might be irrelevant for this case?) So: what is the technique to do this given a function in terms of a usual powerseries, for instance for the geometric series $ g(x)=1+x+x^2+ \ldots $ or some series $ h(x) = K + a*x + b*x^2 + c*x^3 + \ldots $ ?

[edit: minus-sign in f(x) was missing, one numerator in f2 was wrong]

share|cite|improve this question
up vote 3 down vote accepted

Divide the numerator and denominator of $f(x)$ by $x^2$ and set $y=1/x$ then expand for $y$ and you have your expansion at infinity.

share|cite|improve this answer
Ah, thanks! so I also conclude that this is indeed the "expansion at infinity". I also just found some online-links using this term; I'll see whether there is some more general information and whereabouts for this in one of that resources. – Gottfried Helms Nov 23 '10 at 20:31
Yes, that was simply formal polynomial division $ 3x+2x^2 : 1+2x+x^2= 3x - \ldots $ and the other way round $2+3/x : 1+2/x+1/x^2 = 2 - \ldots $ where I was considering convergence w.r.t. x and tried both ways of division. Again thanks, that was helpful – Gottfried Helms Nov 23 '10 at 21:08

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.