Polylogarithm and taylor series

[rylz]

let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series:
Li_n(x)= Ʃ x^k/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...

 

[Dick]

let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series:
Li_n(x)= Ʃ x^k/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 ∞.

 

[rylz]

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?

 

[Dick]

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.