extracting every nth term of a series

Roots of unityMathworldPlanetmath can be used to extract every nth term of a series. This method is due to Simpson [1759].

Theorem. Let ω=e2πi/k be a primitive kth root of unity. If f(x)=j=0ajxj and n0(modk), then


Proof. This is a consequence of the fact that j=0k-1ωjm=0 for m0(modk).

Consider the term involving xr on the right-hand side. It is


If rn(modk), the sum is zero. So the term involving xr is zero unless rn(modk), in which case it is arxr since each element of the sum is 1.

Note that this method is a generalizationPlanetmathPlanetmath of the commonly known trick for extracting alternate terms of a series:


produces the odd terms of f.

Title extracting every nth term of a series
Canonical name ExtractingEveryNmathrmthTermOfASeries
Date of creation 2013-03-22 16:23:34
Last modified on 2013-03-22 16:23:34
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 10
Author rm50 (10146)
Entry type Theorem
Classification msc 11-00