Evaluating the product of two Taylor series

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  • SocoNhydro420
    FFR Veteran
    • Oct 2008
    • 915
    #1

    Evaluating the product of two Taylor series

    I work as an online calculus tutor, and one of the problems that I was helping a student with today seemed a little too difficult to do by brute force. I was wondering if there was a more efficient method, or any other way to solve the problem by hand.

    The problem asks to find the taylor series representation of:

    Ln(x+1)/(1-x)
    About x=0

    Usually, for functions that have nasty derivatives, there are shortcuts in finding their taylor series which the problem is highlighting for practice. For example, to find it for x^2/(1-x^3), you would take the series for 1/(1-x), the geometric series, replace the x's with x^3 and multiply it by x^2.

    Does anyone know how to find an analytic solution using the taylor series of ln(x+1) and/or 1/(1-x)?

    MUST... AAA...

    FMO AAAs (27): Epidermis, Exciting Hyper Highspeed Star, Rottel-da-station, Disconnected Hardkore, Melonmans OP, Battle Theme #37, Fast Asleep, Gacha Gacha Hertz Figu atto Radio, Puzzle, Midnight Dragon, Distorted God, Variations 2, Strangeprogram, Arrogant Cobbler, Kanon Medly ~Metal Wings~, Dance and Zeal, Heavenly Spores, Document 13b, The Divine Suicide of K, Yorukumoryuu Yamikaze, Summer Time Perfume, Chaosmaid, Colorful Course, O (piano version), Ambient Angels, Defection, Jeanie and Caroline
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    • iironiic
      D6 FFR Legacy Player
      FFR Simfile Author
      • Jan 2009
      • 4342
      #2
      Re: Evaluating the product of two Taylor series

      The Taylor series of ln (1+x) is



      And the Taylor series of 1/(1-x) is



      Are you asking for or ? (log x is ln x in WolframAlpha lmfao)

      EDIT: For :



      You would need to multiply the expansion term by term. I like to keep the Maclaurin series of ln x grouped as one factor, and distribute that in the expansion of the geometric series. From this you should get something like:

      +...

      Where the coefficient of x^n is the nth partial sum of the alternating harmonic series.


      EDIT2: For :



      Look at the Maclaurin series of ln(1+x). We can get the Maclaurin series of ln(1-x) by replacing every x with a -x in the series expansion. When we subtract the two series, half of the terms cancel in the expansion. You are left with the series expansion of ln(1+x)-ln(1-x) or . The final answer you should get is:




      Hope this helps (:
      Last edited by iironiic; 05-7-2012, 01:39 AM.

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        • SocoNhydro420
          FFR Veteran
          • Oct 2008
          • 915
          #3
          Re: Evaluating the product of two Taylor series

          That does help, I was thinking that multiplying infinite series out wouldnt be possible, but after some thought i realized that you can solve for the first few coefficients by multiplying and adding together the terms that gave the corresponding order of x, and then finding a pattern.

          Thanks!


          Edit: I was just checkin out your skill determining system, and it looks pretty well thought out. I was wondering about the reason for the square in ( Xp/Q(p) )^2, it seems to me that a linear system would make more sense.

          Heres an EXTREMELY simplified example to show what i mean... suppose theres 10 FGOs in the game, player 1 has AAAd 4 and player 2 has AAAd 6. An estimate of X12 ignoring the almost AAA scores would be 4 and 6 respectively, giving ratings of 12.16 and 12.36. Seems to me that people who have AAAd a little more or less than half of FGOs should have a rating of around 12.5?

          Im thinking the square takes into account the difficulty range of 12's, assuming the AAAs one person has are easier songs than their un-AAAd.

          What is your reason behind it?
          Last edited by SocoNhydro420; 05-8-2012, 07:51 PM. Reason: typo

          MUST... AAA...

          FMO AAAs (27): Epidermis, Exciting Hyper Highspeed Star, Rottel-da-station, Disconnected Hardkore, Melonmans OP, Battle Theme #37, Fast Asleep, Gacha Gacha Hertz Figu atto Radio, Puzzle, Midnight Dragon, Distorted God, Variations 2, Strangeprogram, Arrogant Cobbler, Kanon Medly ~Metal Wings~, Dance and Zeal, Heavenly Spores, Document 13b, The Divine Suicide of K, Yorukumoryuu Yamikaze, Summer Time Perfume, Chaosmaid, Colorful Course, O (piano version), Ambient Angels, Defection, Jeanie and Caroline

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