1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Infinite Sum Question (q-harmonic?)

  1. Mar 20, 2008 #1

    re8

    User Avatar

    I have been working with the following series $\sum_{n=0}^\infty \frac{x^n}{a+x^n}$, where $0<x < 1$. I had a feeling it might be related to q-harmonic series, but I really have no idea:-)

    I am looking for either a good analytic approximation, or even some idea of the sensitivity of this sum to the parameters x and a. But I would grateful if anyone could point me in the right direction.
     
  2. jcsd
  3. Mar 20, 2008 #2

    CRGreathouse

    User Avatar
    Science Advisor
    Homework Helper

    Interesting. The function seems to be well-behaved enough, at least for 'reasonable' values of x and a. In what context did you find this? I wasn't able to find anything in the 'normal places', but I might have missed something.

    What kinds of values do you have for a and x?
     
  4. Mar 20, 2008 #3

    re8

    User Avatar

    Context

    x can be pretty much anything between 0 and 1; I need to check a to see what kinds of restrictions I have on it, but I think it is also between 0 and 1.

    In case you are curious, I'm an economist and it is from a model we formulated:-) It essentially represents the future benefit from varying a certain action x in the model, and we want to show that this future benefit is dominated by the cost today of the action (hence the action is not worthwhile) - hence the need for some idea of the impact of changing x on this sum. I can give you more detail if you like. A bound on the derivative of the sum w.r.t. x would be a start, although these changes aren't necessarily infinitesimal. I tried obvious things like using the initial terms, but it wasn't quite enough.

    I have simulated it for a very fine grid of values for x between 0 and 1 and I get the results I want, but I would like to see if I can get an analytic result.
     
  5. Mar 21, 2008 #4

    CRGreathouse

    User Avatar
    Science Advisor
    Homework Helper

    I don't know of an analytic result. For a fixed value of either x or a it probably wouldn't be hard, but for both... nothing comes to mind.

    How close to the extremes can the values come? Would x = 0.9999 be common? As long x isn't too close to 1 and a isn't too close to 0 it's easy to calculate this.
     
  6. Mar 21, 2008 #5
    I haven't checked this, so I have probably gotten some of the regions of validity wrong, but here are my thoughts on approximating the sum.

    Write $N = \frac{\log a}{\log x}$ so that your series becomes $\sum_{n=0}^\infty \frac{x^n}{a+x^n} = \sum_{n=0}^\infty \frac{1}{1+x^{N-n}}$. Over all integer $n$, this curve is a sigmoid, and $N$ is the knee of the curve (value $1/2$) where terms transition from "near 1" to "near 0"; you can approximate or bound the series separately in these two regimes.

    Things are especially convenient if you can restrict $2N$ to an integer; $\frac{1}{1+x^{N-n}}+\frac{1}{1+x^{N-(2N-n)}}=1$, so terms $n$ and $2N-n$ of the series now sum to $1$ and the sum from $0$ to $2N$ is $\frac{2N+1}{2}$ exactly, and you only have to consider the tail of the sum. But even if $2N$ is not an integer, this is a pretty good approximation for the first $\round{2N}$ terms of the sum (including a fraction of the final term).

    If $1/a$ or $1/x$ is large, then in the tail of the sum ($n>2N$) you can reasonably approximate $\frac{1}{1+x^{N-n}} \approx x^{n-N}-x^{2(n-N)}+\cdots$ using the first few terms of a geometric series; now just sum these to get an estimate of the tail of the series. This estimate isn't very good when $a$ and $x$ are near $1$; but for $x$ near $1$ the terms vary slowly with $n$, and you can approximate the sum by an integral (which evaluates to a hypergeometric function).
     
  7. Mar 22, 2008 #6

    re8

    User Avatar

    Thanks - I'll try this

    Thanks - I'll try that
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Infinite Sum Question (q-harmonic?)
Loading...