# How to estimate $\left(1 + \sqrt{2} + \sqrt{3} + \dots + \sqrt{n} \right) - \frac{2}{3} n \sqrt{n}$?

How do I estimate the error for the sum of reciprocals of square roots. From calculus we know that:

$$\int_0^n \sqrt{x} \, dx = \frac{2}{3} x\sqrt{x} \;\;\Bigg|_{x=0}^{x=n} = \frac{2}{3}n\sqrt{n}$$

I forget the name - midpoint rule, trapezoid rule ?? - basically we want to approximate the integral as a Riemann sum. How do we estimate the error?

$$\left(1 + \sqrt{2} + \sqrt{3} + \dots + \sqrt{n} \right) - \frac{2}{3} n \sqrt{n}$$

To give you a sense of how much we are losing on this approximation, let's draw two pictures.

We are losing all the gray stuff in our approximation, which is quite a lot! I don't really care about the integral, what's important is the difference between all the stuff we are adding and square root of $n$.

The yellow triangle has base $1$ and height $\sqrt{14} - \sqrt{13}$ so the area is:

$$A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 1 \times (\sqrt{\color{#E0E070}{14}} - \sqrt{\color{#E0E070}{13}}) = \frac{1}{2}(\sqrt{n+1} - \sqrt{n}) \approx \frac{1}{4 \sqrt{n}}$$

This suggests the total of all errors is about $\propto \sqrt{\color{#D22}{n}}$ which is not a small amount. Can anyone get the constant of proportionality?

The Euler-Maclaurin machine does crank out such error estimates, but the square root function is not so strange. Can we derive such an estimate in this specific case using basic standard inequalities?

A trapezoidal rule gives a much better approximation: $$\sum_1^n i^{1/2}\approx \frac 1 2(f(1)+f(n))+\int_1^n f(x)\,dx= \frac 1 2 +\frac 1 2 \sqrt n +\frac 2 3 n^{3/2}-\frac 2 3$$ which recovers the biggest component of the error and ends up with an error bound by a small constant. Error analysis can also improve the above estimate to $$\sum_1^n i^{1/2}=\frac 2 3n^{3/2}+\frac 1 2n^{1/2}-\frac 1 6-\frac 1 {24}+\frac 1{24}n^{-1/2}+E_n$$ where non-negative $E_n$ is tiny ($<5\cdot 10^{-4}$).

• Indeed, if we add up the area of the $\color{#20E070}{\text{green}}$ rectangle and the $\color{#E0E070}{\text{yellow}}$ triangle, we do get a trapezium, whose area is $A = \frac{1}{2} \times b \times (h_1 + h_2)$. – cactus314 Nov 16 '15 at 18:03
• @cactus Simple error analysis also shows that simply subtracting $\frac 1 {12}f'(1)=\frac 1{24}$ from the above estimate will recover most of the constant error term (error of less than $4.5\cdot 10^{-4}$ from the correct constant). – A.S. Nov 16 '15 at 18:33
• my question was originally about $\sum \frac{1}{\sqrt{n}}$ but I simplified it to $\sum \sqrt{n}$ after I drew the wrong charts :-P for me the surprise is how accurate I can get my answer without using any calculus at all. there are many result of this kind where I could try to use the same principle. Thank you. – cactus314 Nov 16 '15 at 19:46
• @cactus Can't avoid calculus if you integrate to approximate a sum ;). – A.S. Nov 16 '15 at 20:05
• I am also pointing out a typo in your response... you say $\sum i^{\color{red}{-}1/2}$ you should take out the minus sign. – cactus314 Nov 16 '15 at 20:06

Using the Binomial Theorem, we get \begin{align} k^{3/2}-(k-1)^{3/2} &=k^{3/2}\left[1-\left(1-\frac1k\right)^{3/2}\right]\\ &=k^{3/2}\left[\frac3{2k}-\frac3{8k^2}-\frac1{16k^3}+O\left(\frac1{k^4}\right)\right]\\ &=\frac32k^{1/2}-\frac38k^{-1/2}-\frac1{16}k^{-3/2}+O\left(k^{-5/2}\right)\tag{1} \end{align} and \begin{align} k^{1/2}-(k-1)^{1/2} &=k^{1/2}\left[1-\left(1-\frac1k\right)^{1/2}\right]\\ &=k^{1/2}\left[\frac1{2k}+\frac1{8k^2}+O\left(\frac1{k^3}\right)\right]\\ &=\frac12k^{-1/2}+\frac18k^{-3/2}+O\left(k^{-5/2}\right)\tag{2} \end{align} and \begin{align} k^{-1/2}-(k-1)^{-1/2} &=k^{-1/2}\left[1-\left(1-\frac1k\right)^{-1/2}\right]\\ &=k^{-1/2}\left[-\frac1{2k}+O\left(\frac1{k^2}\right)\right]\\ &=-\frac12k^{-3/2}+O\left(k^{-5/2}\right)\tag{3} \end{align} Combining $(1)$, $(2)$, and $(3)$ gives \begin{align} &\small\left(\frac23k^{3/2}+\frac12k^{1/2}+\frac1{24}k^{-1/2}\right)-\left(\frac23(k-1)^{3/2}+\frac12(k-1)^{1/2}+\frac1{24}(k-1)^{-1/2}\right)\\[6pt] &\small=k^{1/2}+O\left(k^{-5/2}\right)\tag{4} \end{align} Summing, we get that $$\sum_{k=1}^nk^{1/2}=\frac23n^{3/2}+\frac12n^{1/2}+C+\frac1{24}n^{-1/2}+O\left(n^{-3/2}\right)\tag{5}$$ where $C=\zeta\left(-\frac12\right)=-0.207886224977354566\dots$

Why $\boldsymbol{\zeta\left(-\frac12\right)}$?

We can rewrite the reasoning given in this answer, which uses the Euler-Maclaurin Sum Formula, but use the Binomial Theorem, as above. However, the argument is exactly the same. That is, $$\lim_{n\to\infty}\left[\sum_{k=1}^nk^{-z}-\frac1{1-z}n^{1-z}-\frac12n^{-z}\right]$$ converges uniformly to an analytic function for $\mathrm{Re}(z)\gt-1$. For $\mathrm{Re}(z)\gt1$, this function is easily seen to be $\zeta(z)$.

By analytic continuation, we get that this function is $\zeta(z)$ for $\mathrm{Re}(z)\gt-1$. For this question, plug in $z=-\frac12$.

• I've removed the usage of Euler-Maclaurin. – robjohn Nov 16 '15 at 20:59
• This doesn't look right. You've lost much the precision in $(1)$ (for example for $k=1$). How do you go from $(4)$ to $(5)$? You can't sum $O(k^{-5/2})$ for $k=1,..,n$ to get $O(n^{-3/2})$. This reasoning would only apply if we needed $\sum_k^N i^{1/2}$ for large $k$ and hence summed $(4)$ from $k$ to $N$ – A.S. Nov 17 '15 at 0:08
• Where do you think I've lost precision? If $\left|a_k\right|\le Ak^{-5/2}$, then $\left|C-\sum\limits_{k=1}^na_n\right|=\left|\sum\limits_{k=1}^\infty a_n -\sum\limits_{k=1}^na_n\right| \le\frac23An^{-3/2}$. This even agrees with the Euler-Maclaurin result. – robjohn Nov 17 '15 at 0:58
• @A.S.: There are many ways to compute $\zeta(z)$. However, the way that seems most applicable to this question is to extend the approximation above to \begin{align}\small\sum_{k=1}^nk^{1/2} &=\tfrac23n^{3/2} +\tfrac12n^{1/2} +\zeta\left(-\tfrac12\right) +\tfrac1{24}n^{-1/2} -\tfrac1{1920}n^{-5/2} +\tfrac1{9216}n^{-9/2} -\tfrac{11}{163840}n^{-13/2}\\ &\small+\tfrac{65}{786432}n^{-17/2} -\tfrac{223193}{1321205760}n^{-21/2} +\tfrac{52003}{100663296}n^{-25/2} -\tfrac{4741887}{2147483648}n^{-29/2} +O\left(n^{-33/2}\right)\end{align} – robjohn Nov 17 '15 at 7:26
• We are only computing $\sqrt1+\sqrt2+\sqrt3+\dots+\sqrt{10}$ to get $\zeta\left(-\frac12\right)$ to $18$ places. With that, we can compute $\sum\limits_{k=1}^n\sqrt{k}$ very accurately for large $n$. What's the problem with that? – robjohn Nov 17 '15 at 7:41