Alternative proof for $1+2w+3w^2+\ldots+nw^{n-1}= \frac{n}{w-1}$

Question

I want to prove that if $w\neq 1$ is a root of unity, then $$F(w)=1+2w+3w^2+\ldots+nw^{n-1}= \frac{n}{w-1}$$ I have already proved it in two different ways but I am looking for a shorter and more elegant one, using derivatives. I already know that $1+w+w^2+\ldots +w^{n-1}=w+w^2+\ldots+w^n=0$. It turns out that if $$f(w) = w+w^2+\ldots+w^n \quad \mbox{then} \quad f'(w)=F(w)$$ But $f(w)=0$ and therefore $f'(w)=0 \neq F(w)$. Am I missing something? Can't I use derivatives for this argument?

Answer 1

Your development can be understood in two ways:

• $\omega$ is a constant, namely a root of unity. Then taking the derivative on $\omega$ is pointless.

• $\omega$ is a variable, which you will later identify to a root of unity. In this case, $f(\omega)=0$ does not hold yet.

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For an alternative proof, observe that

$$F(\omega)-\omega F(\omega)=1+\omega+\omega^2+\cdots\omega^{n-1}-n\omega^n=-n$$ and

$$F(\omega)=-\frac n{1-\omega}.$$