Boot up your MAPLE system and type:

The system will respond, but not with

or one of its stretched-out forms. Rather it prints the closed form:

This puts a new spin on doing mathematics -- automatic simplification of a formal finite series to a closed-form expression. Clearly this symbolic computing power will accelerate the pace of research in mathematics, pure and applied, but pity the worker who wanted to discover this identity on his own. ("Don't tell me, I want to do it myself!")

MAPLE can be set to show the details of its calculations but in the case of this sum, the system gives no helpful clues to how it found the closed form.

WHY NOT JUST LOOK IT UP?

The infinite series

We can take the derivative (with respect to r), multiply by r and add one (the k=0 term) to arrive at the closed form

This agrees with the limit of our closed form for the finite sum (assuming r<1) as N goes to infinity. Perhaps these techniques could be stretched to produce the closed form in the finite case. But that would still not tell us how the handbook value might be found. The most obvious aproach is backwards, start with the logarithm expression and derive its Fourier series. But we cannot do this unless we know the closed form in advance.

Enough of this! Let us just derive the general case.

DERIVATION OF THE CLOSED FORM

(This is not the slickest way to do it, it is just the first way that I thought of.)

Let   ,

then   .

Using the high-school identity   

we get      

Now multiply by r and split into two sums:

Shift the indices in both sums, to get

Adjust the first and last summands, so that both sums become exact copies of S.

Now simply solve for S (and use the symmetry of cosine):

since we recall that