Summing Over n-th Roots: A Scientific Inquiry

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In summary, the conversation discusses the topic of summing the expression a*r^(1/n) for all n, specifically in relation to symmetric functions and closed form solutions. The conversation also mentions the sum of roots of unity and the roots of other numbers, including how they can be represented as the n'th roots of a real number. In conclusion, the conversation provides various insights and approaches for calculating this sum.
  • #1
imAwinner
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Does anyone know how to sum a*r^(1/n) for all n?
 
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  • #2
Well, if you mean the roots of the equation X^n-r = 0, you should look up symmetric functions.
 
  • #3
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).
 
  • #4
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.
 
  • #5
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
 
  • #6
The sum of roots of unity is zero.
 
  • #7
I know that, what about sums of roots of other numbers?
 
  • #8
The n'th roots of any real number, say r, is [itex]r^{\frac{1}{n}} \zeta_n^k[/itex] where [itex]\zeta_n[/itex] is the primitive nth root of unity. So what will happen when you sum them?

Edit: [tex] 0\leq k \leq n-1 [/tex]
 
  • #9
Thanks Kreizhn! should have noticed that myself =)
 
  • #10
Wait a second, what exactly should I have noticed? I'm summing over n not k.
 

FAQ: Summing Over n-th Roots: A Scientific Inquiry

What is the "Sum of n-th roots"?

The "Sum of n-th roots" is a mathematical concept that involves finding the sum of all the n-th roots of a given number. It is denoted by the symbol ∑√n and is commonly used in algebra and number theory.

How do you calculate the "Sum of n-th roots"?

To calculate the "Sum of n-th roots", you first need to find all the n-th roots of the given number. Then, you simply add up all these roots to get the final sum. For example, if the given number is 8 and n = 2, the n-th roots would be √8 = ±2, and the sum of these roots would be 2 + (-2) = 0.

What is the significance of the "Sum of n-th roots"?

The "Sum of n-th roots" has various applications in mathematics, such as in finding the solutions to polynomial equations. It is also used in complex analysis and number theory to solve problems related to complex numbers and prime numbers, respectively.

Can the "Sum of n-th roots" be negative?

Yes, the "Sum of n-th roots" can be negative. This will depend on the given number and the value of n. For example, if the given number is -8 and n = 3, the n-th roots would be -2, which would result in a negative sum of -2.

Are there any special cases for the "Sum of n-th roots"?

Yes, there are a few special cases for the "Sum of n-th roots". One of them is when n = 1, in which case, the sum of n-th roots would be equal to the given number itself. Another special case is when the given number is 0, in which case, the sum of n-th roots would always be 0 regardless of the value of n.

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