Does this alternating sum of roots converge to $\sqrt2$?

Question

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical lines so that each partition has area $\dfrac{\pi}{p}$. If this background is at all unclear, I would be happy to explain in more depth, but I'll leave all the information relevant to the problem at hand below.

I'm currently interested in the behavior of two sums involving the left endpoint of the rightmost partition; that is, the value of $t$ such that $$2\int_t^1\sqrt{1-x^2}\,\mathrm{d}x=\frac{\pi}{p}$$ Integrating and setting equal to $0$, I obtain this function $$f(t)=\arccos t-t\sqrt{1-t^2}-\frac{\pi}{p}$$

I know $f^{-1}$ exists, as $f$ is monotonic on its domain. However, without an elementary closed form, I'll resort to calling it $F(t)$. I know that for any given $p$, $F$ has exactly one real root, which I'll denote by $F_p$.

The first few terms of $F_p$ are $$\{-1, 0, 0.264932, 0.403973, 0.491862, 0.553293, 0.599061, 0.634705, 0.663385, \ldots\}$$ (Note that $p=1$ means no cuts are made; the disk itself already constitutes a piece.)

Enter the sums. I'm wondering if the following series converge: $$\sum_{p\ge1}F_p\quad\text{and}\quad\sum_{p\ge1}(-1)^pF_p$$ It's easy to see that the first sum diverges, as $\lim\limits_{p\to\infty}F_p=1\neq0$.

I have only numerical evidence to suggest that the alternating sum converges to some value around $1.41067$ based on the sum of the first $200\,000$ roots of $F$. This value, it seems to me, is remarkably close to $\sqrt2$.

In order to establish convergence, I'm thinking the Cauchy criterion for series is my best bet. This would involve showing that for some $k$, I have $$\left|(-1)^{n+1}F_{n+1}+(-1)^{n+2}F_{n+2}+\cdots+(-1)^{n+k}F_{n+k}\right|<\varepsilon$$ for some $\varepsilon>0$ but without any closed form for $F_p$, I'm not sure what I can do here.

My questions are:

• What methods can I employ to prove that the alternating series converges?

• If it converges, what is its value?

• If it indeed converges to $\sqrt2$, is there a neat way of showing this?

The third question would probably be answered by the second, as most interesting convergent sums often involve "neat" methods, but I bring this up because $\sqrt2$ shows up in the unit circle as the length of the chord that makes a central angle of $\dfrac{\pi}{4}$. I'm curious if any connection can be drawn between this fact and the conjectured value of the sum.

Answer 1

Define a sequence $\{F_n\}$ for $n\ge1$ by the equation

\begin{equation} \int_{F_n}^{1}\sqrt{1-x^2}\,dx=\frac{\pi}{n} \end{equation}

Then clearly $\{F_n\}$ in an increasing sequence approaching $1$ as a limit. Therefore the series $\sum_n^\infty F_n$ and $\sum_{n=1}^\infty (-1)^{n}F_n$ both diverge since $\lim_{n\to\infty}F_n\ne0$.

However, consider the following sequences:

The divergent sequence

\begin{equation} s_n=\sum_{k=1}^{n}(-1)^k F_k \end{equation}

and the two subsequences of $\{s_n\}$ defined by $a_n=s_{2n-1}$ and $b_n=s_{2n}$ for $n\ge1$.

The first few terms of $\{s_n\}$ are

\begin{equation} 1, 1, 0.735068, 1.139041, 0.647179, 1.200472, 0.601411, 1.236116, 0.572731 \end{equation}

The first few terms of $\{a_n\}$ are

\begin{equation} 1, 0.735068, 0.647179, 0.601411, 0.572731.\,\ldots \end{equation}

The first few terms of $\{b_n\}$ are \begin{equation} 1, 1.139041, 1.200472, 1.236116,\ldots \end{equation}

So although the sequence $\{s_n\}$ of partial sums of $\sum_{k=1}^{n}(-1)^k F_k$ diverges, it's odd and even termed subsequences $\{a_n\}$ and $\{b_n\}$, decreasing and increasing sequences respectively, possibly converge.

Perhaps it is the sequence $\{b_n\}$ which is suspected to converge to $\sqrt{2}$?

Answer 2

EDIT: See the comment below, this answer is not correct.

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The alternating series definitely converges. A good way to see this is graphically. Below I have used Maple to plot the graph $G$ of the function$$f(t)=2\int_t^1\sqrt{1-x^2}\,dx$$ where $t \in [0,1]$. Moreover, I have also plotted the horizontal lines corresponding to heights $\pi/p$ for $p=2,3,\dots,10$. (Click to enlarge.)

We may write the alternating series as$$\big(F(2)-F(1)\big)+\sum_{k\geq 2}\big(F(2k)-F(2k-1)\big)=1+\sum_{k\geq 2}\big(F(2k)-F(2k-1)\big)$$

Notice that $F(2k)-F(2k-1)$ is always positive; moreover, one can visualize this quantity directly in the image. For instance, for $F(4)-F(3)$, consider the point where $G$ intersects the third and second lines line (from top to bottom). The corresponding $x$-coordinates are respectively $F(4)$ and $F(3)$, so we can take $F(4)-F(3)$ as the length of the interval (in the $x$-axis) with these endpoints. For $F(6)-F(5)$, we can do the same thing replacing the third and second lines by the fifth and fourth lines in the image.

Perhaps the following image will better aid in visualizing the intervals in question; they are highlited black.

Now, of course, these tiny intervals are all disjoint (since $f$ is strictly decreasing) and their total length cannot exceed $1$ (the length of $[0,1]$, the whole domain). In other words, we've shown that the series, as written above, is monotonically increasing and bounded above, so it must be convergent.

I don't have much with respect to other questions right now, but if I come up with something I will edit this.