Complex Formula Proof

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September 19th 2010, 08:55 PM
KatyCar

Complex 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!
September 19th 2010, 09:15 PM
mr fantastic

Quote:

Originally Posted by KatyCar

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!

$S_n = 1 + q + q^2 + ... + q^n$ is a geometric series. The sum is well known. Differentiate both sides.
September 20th 2010, 03:52 AM
HallsofIvy

You could also note that $1- z^n= (1- z)(1+ z+ z^2+ z^3+ \cdot\cdot\cdot\+ z^{n-2}+ z^{n- 1})$ 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.

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