Convergence of a Power Series for Polylogarithm

show boxes

k

1

truncation

5

height

5

modulus

0.5

argument

0.001


absolute value of error = 0.00460551

This Demonstration shows the convergence behavior of the family of power series

∞

∑

n=1

n

z

k

n

on the unit disk

z≤1

for

k=0,1,2…

, which for

z<1

converges to the analytic function

Li

k

(z)

. The graphic on the left shows the graph of a partial sum approximation while the one on the right plots the analytic function that is the limit of the series. The little black dots on both surfaces represent the corresponding images of the same complex number on the unit disc, which you can move around the unit disk by varying its modulus and argument (the lowest two controls). By increasing the truncation parameter you can make the modulus of the error (the difference between

Li

k

(z)

and the approximation) arbitrarily small, except at points on the boundary where the series does not converge.