Polylogarithm and taylor series

rylz
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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 couldn't 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 I am not sure how to prove that...
 
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rylz said:
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 convergence3. when i attempted this part, i couldn't 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 I am not sure how to prove that...


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 ∞.
 
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Dick said:
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 ∞.
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?
 
rylz said:
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?

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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