extracting every term of a series
Roots of unity can be used to extract every term of a series. This method is due to Simpson .
Theorem. Let be a primitive root of unity. If and , then
Proof. This is a consequence of the fact that for .
Consider the term involving on the right-hand side. It is
If , the sum is zero. So the term involving is zero unless , in which case it is since each element of the sum is .
Note that this method is a generalization of the commonly known trick for extracting alternate terms of a series:
produces the odd terms of .
|Title||extracting every term of a series|
|Date of creation||2013-03-22 16:23:34|
|Last modified on||2013-03-22 16:23:34|
|Last modified by||rm50 (10146)|