Summing Over n-th Roots: A Scientific Inquiry

[imAwinner]

Does anyone know how to sum a*r^(1/n) for all n?

 

[robert Ihnot]

Well, if you mean the roots of the equation X^n-r = 0, you should look up symmetric functions.

 

[imAwinner]

I can't see how symmetric functions would help. I'm looking for a closed form solution for the given sum, in the sense that the infinite sum of a*r^n = a*(1-r^(n+1))/(1-r), I'm looking for the infinite sum of a*r^(1/n).

 

[Hurkyl]

Could you write what you mean, rather than abbreviating it? I can't tell precisely what you mean, and my best guesses for what you mean are very obviously not convergent sums.

 

[imAwinner]

What does \sum_{k=0}^{n} a*r^(1/k) equal? Given that |r| < 1, a and r are constants.
In the sense that the geometric progression \sum_{k=0}^{n} a*r^k equals a*(1-r^(n+1))/(1-r).

Cheers

 

[Kummer]

The sum of roots of unity is zero.

 

[imAwinner]

I know that, what about sums of roots of other numbers?

 

[Kreizhn]

The n'th roots of any real number, say r, is r^{\frac{1}{n}} \zeta_n^k where \zeta_n is the primitive nth root of unity. So what will happen when you sum them?

Edit: 0\leq k \leq n-1

 

[imAwinner]

Thanks Kreizhn! should have noticed that myself =)

 

[imAwinner]

Wait a second, what exactly should I have noticed? I'm summing over n not k.