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Polylogarithm and taylor series

  1. Dec 6, 2012 #1
    let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series:
    Lin(x)= Ʃ xk/kn



    1.calculate the radius of convergence


    3. when i attempted this part, i couldnt use theratio or root test, so by comparison i got R=∞

    2. Prove that (1-x)2 Li-1 (x)= x

    im not sure how to go about this. i know that Li-1 (x)= 1/(1-x)2 but im not sure how to prove that..........
     
  2. jcsd
  3. Dec 6, 2012 #2

    Dick

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    For the first part, I have no idea what you compared with. For the second just look at the taylor series expansion of 1/(1-x)^2. If that's Li-1(x), and it is, it certainly doesn't have radius of convergence ∞.
     
    Last edited: Dec 6, 2012
  4. Dec 8, 2012 #3
    hey! so i sorted out the first part but how do i axctually prove that Li-1 (x) is equal to 1/(1-x)^2?
     
  5. Dec 8, 2012 #4

    Dick

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    I told you. Find the taylor series expansion of 1/(1-x)^2. Compare it with the series definition of your polylogarithm.
     
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