Converting a Product Function to a Summation
Date: 07/24/2004 at 03:25:12 From: Jay Subject: convert product to summation How do I convert the product of n terms to a summation? For example, if f(x) = a(x - a1)(x - a2)...(x - an), how do I get f(x) = the sum of some series?
Date: 07/24/2004 at 10:43:06 From: Doctor Fenton Subject: Re: convert product to summation Hi Jay, Thanks for writing to Dr. Math. You would need to compute what are called the "elementary symmetric functions" of the roots a1,...,an. If we define sigma_1(x(1),x(2),...,x(n)) = x(1) + x(2) + ... + x(n) sigma_2(x(1),...,x(n)) = x(1)x(2) + x(1)x(3) + ... + x(n-1)x(n) (you take all possible products of the form x(i)x(j), with i < j, and add: for n=4, you would have x1x2 + x1x3 + x1x4 + x2x3 + x2x4 + x3x4, for example) sigma_3(x(1),...,x(n)) = sum of all products x(i)x(j)x(k) , with i < j < k ; and so on, then (x - r(1))(x - r(2))...(x - r(n)) is equal to the sum n --- \ x^n + / (-1)^i sigma_i(r(1),r(2),...,r(n))x^i . --- i=1 For example, if your f(x) is a cubic, then (x - a1)(x - a2)(x - a3) = x^3 - (a1 + a2 + a3)x^2 + (a1a2 + a1a3 + a2a3)x - a1a2a3 If you have any questions or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
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