Formula to Sum a Series of Square RootsDate: 07/03/2004 at 06:56:19 From: Ng Subject: Sum of the roots of x I know the formulas for the sums of x, x^2, and x^3 but is there a way to find the sum of x^(1/2)? I'm referring to this situation: n Sigma r^(1/2) r=1 Date: 07/03/2004 at 17:59:08 From: Doctor Vogler Subject: Re: Sum of the roots of x Hi Ng, Thanks for writing to Dr Math. The short answer is that there is not a simple closed-form formula for the sum you want. However, there is an approximate formula, which is: n If S = sum r^(1/2), then S is no more than r=1 (2/3)n^(3/2) + (1/2)n^(1/2) - 1/6, but no less than (2/3)n^(3/2) + (1/2)n^(1/2) + 1/3 - (1/2)2^(1/2), and this gives you S with an error of less than 0.21. Where did I get this formula? Here: When trying to approximate the sum of a simple function, the following theorems are very useful: If f > 0 is twice differentiable and for all x >= 1 we have (1) f' <= 0 : f(n) <= sum f(k) - int f(x) dx <= f(1) (2) f" <= 0 : f(1) - f(2)/2 <= sum f(k) - int f(x) dx - f(n)/2 <= f(1)/2 (3) f' >= 0, f" >= 0 : 0 <= sum f(k) - int f(x) dx - (1/2)(f(n) + f(1)) <= f'(n)/4 where all sums are from k=1 to k=n, and all integrals are from x=1 to x=n. If you have any questions about this 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 Vogler, The Math Forum http://mathforum.org/dr.math/ |
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