Closed expression for sum $\sum_{k=1}^{\infty} (-1)^{k+1}\frac{\left\lfloor \sqrt{k}\right\rfloor}{k}$

Question

Inspired by the recent question if the series $\sum_{k=1}^{\infty} \frac{\sqrt{k}-\left\lfloor \sqrt{k}\right\rfloor}{k}$ diverges (which is the case) I became interested in the alternating series which is convergent by the Leibniz criterion.

The core of the problem is then the question if this sum

$$s = \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\left\lfloor \sqrt{k}\right\rfloor}{k}\simeq 0.591561$$

has a closed expression. Here $\left\lfloor {x}\right\rfloor$ is the greatest integer less than or equal to $x$.

I have found a nice integral representation for $s$ but I could not find a closed expression. Also, due to the slow convergence of the sum it is not trivial to get a numerical result with high accuracy which might be necessary to identify a possible closed expression.

Problems

a) find a closed expression for $s$
b) find the numerical result exact to 20 decimal places

Answer 1

Results

I have not found a closed form of $s$. However, I will show below that the sum

$$s = \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\left\lfloor \sqrt{k}\right\rfloor}{k}\tag{1}$$

has the following integral representation

$$s_i = \int_0^1 f(x) \, dx\tag{2a}$$

where the integrand is defined as

$$f(x) = \frac{1-\vartheta _4(0,x)}{2 x (x+1)}\tag{2b} $$

Here

$$\vartheta _4(u,q) = 1 + 2 \sum_{n=1}^{\infty} (-1)^n q^{n^{2}} \cos(2 n u)\tag{3}$$

is a Jacobi theta function $[1]$.

The integrand of $s_i$ looks pretty harmless

Derivation

We start by writing down a list of summands of $s$ long enough to see a pattern

$$s\simeq \left\{1,-\frac{1}{2},\frac{1}{3},-\frac{2}{4},\frac{2}{5},-\frac{2}{6},\frac{2}{7},-\frac{2}{8},\frac{3}{9},-\frac{3}{10},\frac{3}{11},-\frac{3}{12},\frac{3}{13},-\frac{3}{14},\frac{3}{15},-\frac{4}{16},\frac{4}{17},-\frac{4}{18}\right\}$$

We see that the list can be decomposed into sublists

$$s_1= \left\{1,-\frac{1}{2},\frac{1}{3}\right\}$$

$$s_2= \left\{-\frac{2}{4},\frac{2}{5},-\frac{2}{6},\frac{2}{7},-\frac{2}{8}\right\}= 2 \left\{-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\frac{1}{7},-\frac{1}{8}\right\}$$

$$s_3= \left\{\frac{3}{9},-\frac{3}{10},\frac{3}{11},-\frac{3}{12},\frac{3}{13},-\frac{3}{14},\frac{3}{15}\right\}= 3 \left\{\frac{1}{9},-\frac{1}{10},\frac{1}{11},-\frac{1}{12},\frac{1}{13},-\frac{1}{14},\frac{1}{15}\right\}$$

Notice that the denominators of the sublist $s_1$ runs from $1$ to $3$, of $s_2$ from $4$ to $8$,of $s_3$ from $9$ to $15$ resp., in general of sublist $s_m$ from $m^2$ to $(m+1)^2-1=m(m+2)$.

To express the pattern formally we use the alternating harmonic sum defined as

$$A(n) = \sum _{k=1}^n \frac{(-1)^{k+1}}{k}\tag{4}$$

Then we can write

$$s_1 = A(3),\\ s_2 = 2 (A(8) -A(3)),\\ s_3 = 3(A(15) - A(8)) $$

and for the partial sums

$$p_1 = s_1 = A(3), \\p_2 = s_1+s_2 = 2 A(8) -A(3), \\p_3 = p_2+s_3 = 3 A(15) - A(8)-A(3)$$

the general partial sum of index $m$ is then

$$p_{m} = m A((m+1)^2-1) - \sum_{k=2}^{m} A(k^2-1)\tag{5}$$

Observing now that

$$A(n) = \sum_{k=1}^n (-1)^{k+1}\int_0^1 x^{k-1}\,dx= \int_0^1 \sum_{k=1}^n (-1)^{k+1} x^{k-1}\,dx= \int_0^1 \frac{1-(-1)^n x^n}{x+1} \, dx\tag{6}$$

we get

$$p_{m} =\int_0^1 \left( \frac{m \left((-1)^{m (m+2)} x^{m (m+2)-1}+1\right)}{x+1}-\sum _{k=2}^m \frac{(-1)^{k^2} x^{k^2-1}+1}{x+1}\right) \,dx\tag{7}$$

Without changing the value we can extend the second sum down to $k=1$. Now we observe that the contribution $\frac{m}{1+x}$ cancels out and that the parity of $k^2$ is the same as that of $k$ and similarly $m(m+2) \sim m$ so that we have

$$p_{m} = \int_0^1 \left(\frac{m \left((-1)^{m} x^{m (m+2)}\right)}{x(x+1)}-\sum _{k=1}^m \frac{(-1)^{k} x^{k^2}}{x(x+1)}\right) \,dx\tag{8}$$

Now we need the limit $m\to\infty$ to get $s=\lim_{m\to \infty } \, p_{m}$.

The first integral is given by

$$I_1(m) = m (-1)^m \int_0^1 \frac{x^{m (m+2)}}{x (x+1)} \, dx\\=\frac{1}{2} (-1)^m m \left(\psi ^{(0)}\left(\frac{1}{2} (m+1)^2\right)-\psi ^{(0)}\left(\frac{1}{2} m (m+2)\right)\right)\tag{9}$$

For $m>>1$ we find that $I_1 \sim \frac{(-1)^m}{2 m}$ so that it vanishes in the limit.

For the limit of the second integral

$$I_2(m) = -\int_0^1 \sum _{k=1}^m \frac{(-1)^{k} x^{k^2}}{x(x+1)} \,dx\tag{10}$$

we have to calculate the sum of the integrand up to $m\to\infty$. Observing $(3)$ we obtain $(2)$. QED.

Discussion

0. Honestly speaking, I didn't expect to find an integral representation because I thought that sharply discontinuous aggregates like $\left\lfloor x\right\rfloor$ would not lead to a smooth formula. But, luckily, my feelings turned out to be misleading, and I was pushed forward by the rather straightforward derivation itself.

1. Series expansion of the integrand

The list of terms of the series expansion of the integrand starts like this

$$f(x) = \left\{1,-x,x^2,-2 x^3,2 x^4,-2 x^5,2 x^6,-2 x^7,3 x^8,-3 x^9,3 x^{10},-3 x^{11},3 x^{12},-3 x^{13},3 x^{14},-4 x^{15}\right\}$$

When integrated

$$s=\int_0^1 f(x) \,dx \simeq \left\{1,-\frac{1}{2},\frac{1}{3},-\frac{1}{2},\frac{2}{5},-\frac{1}{3},\frac{2}{7},-\frac{1}{4},\frac{1}{3},-\frac{3}{10},\frac{3}{11},-\frac{1}{4},\frac{3}{13},-\frac{3}{14},\frac{1}{5},-\frac{1}{4}\right\}$$

we get back where we started from.

This gives me some comfort because I felt a little uneasy about the general validity of the limit while selecting the special form of the partial sums

2. We have in fact found an integral representation also for the sum

$$h = \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\sqrt{k}-\left\lfloor \sqrt{k}\right\rfloor}{k}$$

because the trivial part is

$$\sum _{k=1}^{\infty } \frac{(-1)^{k+1}}{\sqrt{k}}=-\left(\sqrt{2}-1\right) \zeta \left(\frac{1}{2}\right)\simeq 0.604899$$

3. Accuracy

The accuracy issue is briefly that - roughly speaking - Mathematica gives a different numerical result for the integral with NIntegrate than for the sum with NSum. I believe that the value obtained with NIntegrate is better because the integrand is alsmost trivial (see the graph). We had a similar topic recently here.

In the meantime, Yuriy S in a comment has given this numerical value for the integral $(2)$ with Mathematica's NIntegrate, and WorkingPrecision -> 30

$$i_{Yuriy} = 0.591560779349817340213846903345$$

I can confirm this result.

I have calculated the sum with NSum and different values of WorkingPrecision. The results are wobbling appreciably about the limiting value as can be seen in the picture

And I can ony give this very modest result (the average)

$$s_{WH,NSum} = 0.59123$$

Alternatively, the plain Sum of the first million terms is

$$s_{WH,Sum} = 0.5910$$

The accuracy is but $\frac{1}{\sqrt{k_{max}}} \simeq 10^{-3}$

River Li in his solution $[2]$ has transformed the limiting form of the sum $(5)$ into the better converging double sum

$$s_{RL} = \log(2) + \sum_{n=1}^\infty (-1)^n n \sum_{i=1}^{n} \frac{1}{(n^2 + 2i-1)(n^2+2i)}$$

The n-summand has a closed form in terms of polygamma functions and goes asymptotically like $\frac{1}{n^2}$. Hence the convergence is similar to that of Dirichlet $\eta(2)$.

Mathematica finds 5 valid digits for $s_{RL}$ with 1000 n-summands in a few seconds, but refuses to sum 2000 terms is acceptable time.

However, River Li found twenty digits using Maple, a result he later confirmed using the "Convergence Acceleration methods for Alternating Sums" as described here $[3]$ with only 28 terms. This method claims that you can get high precision results from just a few tens of terms. Usage of the method is nicely described in the update of River Li's solution.

Hence I conclude that using summation to find the value of the sum with high accuracy as requested in problem b) needs sophisticated summation methods which provide convergence acceleration, accompanied by a good SC-tool.

Here we are lucky to have the integral representation of the sum for which Mathematica delivers as many digits as requested.

4. Generalization

If we consider the similar problem with the p-th root instead of the square root we have

$$s(p) = \sum _{k=1}^{\infty } \frac{(-1)^{k+1} \left\lfloor k^{1/p}\right\rfloor }{k} = \int_{0}^{1} f(p,x)\,dx$$

where now the integrand is given by

$$f(p,x)=\frac{\sum _{m=1}^{\infty } (-1)^{m-1} x^{m^p}}{x (x+1)}$$

I know no name for this special function which replaces the Jacobi theta function.

The problem with a rational exponent $\frac{p}{q}$ with $1 \lt p\lt q$ seems to be much more difficult to tackle.

5. Using Fourier expansion

We can get rid of the floor function using the Fourier series

$$\left\lfloor x\right\rfloor = x -\frac{1}{2} + \frac{1}{\pi}\sum_{k=1}^{\infty} \frac{\sin(2 \pi k x)}{k}\tag{5.1}$$

@Jam has pursued this approach in https://math.stackexchange.com/a/3452471/198592, and ended with this sum to be evaluated

$$d=-\frac{1}{\pi}{\sum_{n\geq 1}\sum_{k\geq 1}\frac{\left(-1\right)^{n}\sin\left(2\pi k\sqrt{n}\right)}{nk}}\tag{5.2}$$

We can do the $k$-sum

$$\sum_{k\geq 1} \frac{\sin\left(2\pi k\sqrt{n}\right)}{k}= \frac{i}{2}\left(\log \left(1-e^{2 i \pi \sqrt{n}}\right)-\log \left(1-e^{-2 i \pi \sqrt{n}}\right)\right) \\ = \frac{1}{2} i \log \left(\frac{1-e^{2 i \pi \sqrt{n}}}{1-e^{-2 i \pi \sqrt{n}}}\right)\\ =\frac{1}{2} i \log \left(-e^{2 i \pi \sqrt{n}}\right)\tag{5.3}$$

Here is another expression for d

$$d_{nsq}=\frac{i}{2\pi }\sum _{m=1}^{\infty }(-1)^m \sum _{j=1}^{2 m} (-1)^j\frac{ \log \left(-e^{2 i \pi \sqrt{j+m^2}}\right)}{ \left(j+m^2\right)}\tag{5.10}$$

The convergence is satisfactory as can be seen from the following graph

References

$[1]$ http://mathworld.wolfram.com/JacobiThetaFunctions.html
$[2]$ https://math.stackexchange.com/a/3450665/198592
$[3]$ https://people.mpim-bonn.mpg.de/zagier/files/exp-math-9/fulltext.pdf

Answer 2

Update

We may use the convergence acceleration of alternating series developed by Cohen, Villegas, and Zagier. Let $$s = \ln 2 + \sum_{n=1}^\infty (-1)^n n \sum_{i=1}^{n} \frac{1}{(n^2 + 2i-1)(n^2+2i)}$$ and $$s_n = \ln 2 + \sum_{k=1}^n \frac{c_{n,k}}{d_n}\sum_{i=1}^k \frac{k}{(k^2 + 2i-1)(k^2+2i)}$$ where \begin{align} d_n &= \frac{(3+\sqrt{8})^n + (3-\sqrt{8})^n}{2}, \\ c_{n,k} &= (-1)^k \sum_{m=k+1}^n \frac{n}{n+m} \binom{n+m}{2m} 2^{2m}. \end{align} From Proposition 1 in [1], we have $$|s-s_n| \le \frac{s}{d_n}.$$

Maple: $\mathrm{evalf}(s, 30) = 0.591560779349817340213846903345$, $\mathrm{evalf}(s_{28} - s, 30) = 1.6944769437\cdot 10^{-21}.$

[1] Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, "Convergence Acceleration of Alternating Series".

Previously written

We have \begin{align} s &= \sum_{k=1} (-1)^{k+1} \frac{\lfloor \sqrt{k}\rfloor}{k}\\ &= \sum_{n=1}^\infty \left(\sum_{k = n^2} (-1)^{k+1} \frac{\lfloor \sqrt{k}\rfloor}{k}\right) + \sum_{n=1}^\infty \left(\sum_{n^2 < k < (n+1)^2} (-1)^{k+1} \frac{\lfloor \sqrt{k}\rfloor}{k}\right)\\ &= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} - \sum_{n=1}^\infty (-1)^n n \sum_{i=1}^{2n} (-1)^i \frac{1}{n^2 + i}\\ &= \ln 2 - \sum_{n=1}^\infty (-1)^n n \sum_{i=1}^{2n} (-1)^{i} \frac{1}{n^2 + i}\\ &= \ln 2 + \sum_{n=1}^\infty (-1)^n n \sum_{i=1}^{n} \frac{1}{(n^2 + 2i-1)(n^2+2i)}.\tag{1} \end{align} Maple can give numerical approximation of (1) with high accuracy. Or we may use "Convergence Acceleration of Alternating Series" technique to calculate (1).