Walker's Equation Validation


Thank you.  My research on this equation has been educational.  I am attaching what I found out.

I suspect that the sum of sums is original with you.  I have spelled it out as two sums in equation #1.  Switching the two sums (equation #2) just gives us all the same terms, in a different order (in your attachment, the rows become columns and vice versa).  The right sum of equation #2 is a geometric series, which I have evaluated (using the formula for the sum of a geometric series) in #3, and simplified in #4.  Equation #4 is a well known series, which evaluates to 1 (equations 5, 6, and 7 show how to evaluate it).  Surprisingly, equation #5 is equivalent to equation #4 (perform the subtraction in #5 by calculating a common denominator to see that they are the same).  Writing out this series in #6, we see that all of the terms subtract out except the first term (one).  So the sum is 1. 

Jim Loy