Approximation to an average of integer square roots

UnitClash_.variable---.--UnitClash_.rangestart-*--In summary, a user has discovered an approximation for the average of integer square roots, which has a relative error of about 5% and is asymptotic as n tends towards infinity. This approximation can also be used to find the approximate minimum of a function involving tanh and cosh. The conversation also mentions a more accurate approximation using calculus.
  • #1
phasic
21
0
I have stumbled upon an approximation to the average of integer square roots.

[itex]\sum^{n}_{k=1}{\sqrt{k}/n} \approx \sqrt{median(1,2,...,n)} [/itex]

Sorry I am not very good at LaTeX, but I hope this comes across okay. Could anyone explain why this might be happening?

In fact, I just discovered that the error increases in the shape of the sqrt function and is only off by about 4 at n = 10000, about a 5% error. Interestingly, this relative error is asymptotic as n tends towards infinity and may converge? If that were the case, this approximation will always have around 5% relative error...?

This had an interesting consequence on finding where Ʃ(tanh(x-sqrt(k))) = 0. It turns out if x is the average of the square roots up to n, since tanh is an odd function, the sum will be close to 0. Indeed, the approximate minimum at the average is close to the numerically calculated minimum of Ʃ log(cosh(x-sqrt(k)))

I hope this makes sense! Any thoughts are welcome.
 
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  • #2
Do you know any calculus? That is related to something called a Riemann sum.
In calculus one learns that
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n \sqrt{\frac{k}{n^3}}=\frac{2}{3}$$
So we can see that when n is large
$$\frac{1}{n} \sum_{k=1}^n \sqrt{k} \sim \frac{2}{3}\sqrt{n}$$
A more accurate approximation if you are interested is
$$\frac{1}{n} \sum_{k=1}^n \sqrt{k} \sim \frac{2}{3} \sqrt{n} - \frac{1}{2 \sqrt{n}}+\frac{1}{n} \zeta \left( -\frac{1}{2} \right) + \frac{1}{24n \sqrt{n}}$$
where
$$\zeta \left( -\frac{1}{2}\right) \sim -0.20788622497735456601730672539704930222626853128767253761011...$$
http://www.wolframalpha.com/input/?....**Integral.rangestart-.*Integral.rangeend---
 

1. What is "Approximation to an average of integer square roots"?

"Approximation to an average of integer square roots" is a mathematical concept used to estimate the average of the square roots of a set of whole numbers. It involves finding the square root of each number, adding them together, and dividing the sum by the total number of numbers in the set.

2. Why is "Approximation to an average of integer square roots" useful?

This concept is useful in situations where we want to get a general idea of the square root of a set of numbers without calculating the exact values. It can also be used to compare the square roots of different sets of numbers.

3. How is "Approximation to an average of integer square roots" calculated?

To calculate the approximation, we first find the square root of each integer in the set using a calculator or by hand. Then, we add all the square roots together and divide the sum by the total number of integers in the set.

4. Is the approximation always accurate?

No, the approximation to an average of integer square roots is not always accurate. It is an estimate and may not give the exact value of the average square root. However, the more numbers included in the set, the more accurate the approximation will be.

5. Can "Approximation to an average of integer square roots" be used for non-integer numbers?

No, this concept is specifically for finding the average of integer square roots. It cannot be used for non-integer numbers, as the square root of non-integer numbers will also be non-integer and cannot be added to the sum.

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