Find a non-summation expression for the value of the sum .
To find the sum , we first note that , so that our sum . Next, we look back at this post, where it is demonstrated that
Using a similar method, we see that
And thus, for ,
which, using the quotient rule, gives us
However, we can express this more compactly using trigonometric identities. With the product-to-sum formulas, we see:
Substituting these into the numerator, and expanding, we see
which is much more compact.
Lastly, one could also use the half-angle identity to give equivalent form
Now, when x is an integer multiple of 2π. For these values, since n is an integer, , and so our expression has the indeterminate form 0/0. Thus, we can apply l’Hôpital’s rule (twice) for the limit as x approaches zero (and, thus due to the period, any of the other singularities):
so the singularities are removable, with a limit which matches the value expected from the sum:
for x=0, .