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Find the general expression from k=0 to 'n'

  1. Jan 1, 2012 #1
    Hi guys... this problem is really annoying me

    How to find:
    [tex]\displaystyle{\sum_{k=0}^{n}\frac{1}{2^k+3^k}}[/tex]

    I can clearly see that it converges

    [tex]\frac{1}{3^k+3^k}<\frac{1}{2^k+3^k}<\frac{1}{2^k+2^k}[/tex]

    [tex]\sum_{k=0}^{\infty}\frac{1}{3^k+3^k}<\sum_{k=0}^{\infty}\frac{1}{2^k+3^k}<\sum_{k=0}^{\infty}\frac{1}{2^k+2^k}[/tex]

    [tex]\frac{1}{2}\sum_{k=0}^{\infty}\frac{1}{3^k}<\sum_{k=0}^{\infty}\frac{1}{2^k+3^k}<\frac{1}{2}\sum_{k=0}^{\infty}\frac{1}{2^k}[/tex]

    [tex]\frac{1}{2}\frac{1}{1-1/3}<\sum_{k=0}^{\infty}\frac{1}{2^k+3^k}<\frac{1}{2}\frac{1}{1-1/2}[/tex]

    [tex]\frac{3}{4}<\sum_{k=0}^{\infty}\frac{1}{2^k+3^k}<1[/tex]

    But how do I find the general expression from k=0 to 'n' or even the exact value as n goes to infinity?

    Thanks in advance
     
  2. jcsd
  3. Jan 1, 2012 #2
    Re: sum

    You must be assuming that it does. I bet there is no closed form.
    It could be a "wild goose chase."

    So, the origin of the problem and/or who presented it
    need to be considered.
     
  4. Jan 2, 2012 #3
    Re: sum

    What do you mean by "wild goose chase."?

    I was wondering if is there any way of separating [itex]\frac{1}{2^k+3^k}[/itex] in two or more simple fractions...
     
  5. Jan 2, 2012 #4
    Re: sum


    Here might be the Devil's advocate:

    It would be a wild goose chase, in fact, there is not a solution.
    Looking for something that turns out to not to be there
    would be a wild goose chase. It is for the consideration of
    users here not working on a problem if there is not a definite solution,
    but that there was supposed to be.
     
    Last edited: Jan 2, 2012
  6. Jan 3, 2012 #5
    Re: sum

    OK but I must confess that I love wild goose hunting, so I was wondering if someone could just give me a tip on how shall I load the rifle. And if I hit the goose, I'll seve you with one of the best and popular portuguese dishes "Rice with geese"
     
  7. Jan 3, 2012 #6
    Re: sum

    The size of rifle should be not for goose hunting, but rather for elephant and better for dragon !
    You have a small chance to find a closed form for this kind of infinite series if you are very smart in using special functions and especially if you have a perfect knowledge of the q-polygamma functions (do not confuse with the usual polygamma functions).
    Nevertheless, I am quite certain that the closed form can be expressed in terms of q-polygamma functions.
    But instead of spending a lot of time in doing it, I have the pleasure to let the work to be done by somone else more available and more smart than myself. :devil:
     
  8. Jan 3, 2012 #7
    Re: sum

    I'm sorry my alleged arrogance, it was never my intention to state that I'm more smart than anyone, I was just trying to find a solution to a problem which I found thrilling.
    Q-polygamma functions might be the answer.

    Thank you very much for your kind attention
     
  9. Jan 3, 2012 #8
    Re: sum

    I never intended to call someone arrogant.
    Simply, I was in the mood to joke.
    Have a nice time with the q-polygamma ! :rofl:
     
  10. Jan 3, 2012 #9
    Re: sum

    Thank you very much... For sure I'll have :)
     
  11. Jan 4, 2012 #10
    Re: sum

    Hello joao_pimentel !

    finally, I went to the goose hunting.
    With the invavuable help of WolframAlpha for the most boring part of job, I don't comme back with empty bag.
    In the joint page, find the formulas for finite and infinite series with general term 1/(a^k+b^k).
    Numerical verification in case a=3, b=2 . (More digits could be provided, thanks to Wolframalpha). Of course, the direct numerical computation of the sum is much easier than with these complicated formulas.
     

    Attached Files:

  12. Jan 4, 2012 #11
    Re: sum

    God bless JJacquelin

    Outstanding results I must confess, for sure my rifle would never be enough powerful to shot down this dragon.

    Thank you very much for your work and for your kindness!

    I read it carefully and I must confess I enjoyed this "Rice with geese à lá JJacquelin"

    Thanks again, really
     
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