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.

Printable View

- June 20th 2012, 02:07 AMgregx22Root of unity sum
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. - June 20th 2012, 03:00 AMsimamuraRe: Root of unity sum
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 - June 20th 2012, 06:56 AMSorobanRe: Root of unity sum
Hello, gregx22!

Quote:

. .