Suppose q = e^[(2*pi*i)/n]

Show that:

1 + 2*q + 3*q^2 + ... + n*q^(n-1) = n/(q-1)

After testing with various n I have found that this is true, however I am having trouble with the general proof.

Thanks!

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- Sep 19th 2010, 08:55 PMKatyCarComplex Formula Proof
Suppose q = e^[(2*pi*i)/n]

Show that:

1 + 2*q + 3*q^2 + ... + n*q^(n-1) = n/(q-1)

After testing with various n I have found that this is true, however I am having trouble with the general proof.

Thanks! - Sep 19th 2010, 09:15 PMmr fantastic
- Sep 20th 2010, 03:52 AMHallsofIvy
You could also note that $\displaystyle 1- z^n= (1- z)(1+ z+ z^2+ z^3+ \cdot\cdot\cdot\+ z^{n-2}+ z^{n- 1})$ to get a similar result to the "geometric series". Of course, it is recognizing that you can differentiate term by term that is the crucial step.