Approximating square roots | Population Algorithms by Petar Maymounkov

Approximating square roots - 2

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Approximate $\sqrt{1} + \sqrt{2} + \cdots + \sqrt{50}$ in your head (or using a sheet of paper or two) as best as you can. Anonymous.

#Shubham at 4:27 am on June 25th, 2008: My friend lini did the following: Between n^2 and (n+1)^2, there are exactly 2*n numbers. Lets approximate the square root of first n of these by n and the square root of the last n of these as (n+1). This gives the following series:
1*2 + 2*4 + 3*6 + 4*8 + 5*10 + 6*12 + 7*8
which gives 238 ( the real sum is 239.03580060352076 )
Is there a better quick approximation trick??
#petar at 4:11 pm on June 26th, 2008: This problem has no “right” answer obviously, unless you come up with a way to approximate to arbitrary precision that is easy to explain. I.e. you can’t say “find Taylor expansion of square root function” or “do FFT on square root function”. That said, your solution is very nice. Another acceptable solution is to compute the continuous integral of x^{1/2} e.g.