Let X be one of the nth roots of unity.
Find the sum:
I've been trying to figure this out but nothing...
I know that sum of roots of unity is 0. But I have no clue about this one.
Let
This is geometric progression with first member x and q=x.
Therefore, its sum is
We obtained equality:
Differentiating with respect to x yields:
Since x is n-th root of unity then and we can write above equality as
Or
Again, since then
Finally,
As can be seen from formula x=1 can't be applied to the formula.
When x=1
Therefore, if x=1 , otherwise