Formula to Sum a Series of Square Roots

Date: 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/