A geometric progression of form:
Can be summed from N0 to N1 inclusive as follows:
Subtracting the second equation from the first yields:
In particular, if N0=0, N1=N-1, this reduces to:
If a is of form:
then we have:
If we look at the pure real (fraction) in the above expression, we can observe that the numerator is zero for any integral value of x. The denominator is zero for any x which is an integral multiple of N. In the cases where x is an integral multiple of N, both the numerator and denominator are zero, but application of L'Hopitals rule tells us that the fraction has magnitude N. The fraction is zero for all other integral x.
So, to recap, as far as integral values of x are concerned, the sum is zero unless x is also multiple of N (including zero), in which case the magnitude of the sum is N. In fact, by considering distinct cases of N odd and N even, it is easy to show than the sum is always +N if x is a multiple of N. (This is not altogether surprising if we recall that original form of the expression as a sum of complex exponentials.)
In Annex A, we were presented with an expression of form:
This can be related to our result by associating..
Now both m and n range over 0..N-1, so..
Therefore, we need not consider any other integral multiples of N other than x=0, (n=m).
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