An “original” proof of the Basel Sum

I recently got bored and decided to try to find a proof of the Basel Sum. In an unexpected departure from tradition, I actually found one. As a further surprise, it seems to be a lot simpler than any other proof I’ve seen: just a few steps, and each one pretty obvious.

A reminder: the Basel sum states that

    \begin{equation*} \sum_{1}^{\infty}\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots = \pi^2/6 \end{equation*}

And an important clarification: this proof is “original”, in the sense of I came up with it myself, yay, but most definitely not original in the sense of I was the first to come up with this proof. The Basel sum is exceedingly well traveled territory mathematically, and I would be surprised if I was in the first 1000 people to come up with this proof independently.

Before we start the proof, a reminder of the definition of the double factorial. For an even number 2k, 2k!! = 2 \times 4 \times 6 \ldots \times (2k). Similarly, for an odd number (2k+1), (2k+1)!! = 1 \times 3 \times 5 \ldots \times (2k+1). And we can choose to define (-1)!! = 0!! = 1.

Step 1:

Let

    \[S = \sum_{1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots\]

Then,

    \[\frac{S}{4} = \frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \ldots\]

Subtracting these two equations, we get

(1)   \begin{equation*} \boxed{\frac{3S}{4} = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \ldots} \end{equation*}

Step 2:

(2)   \begin{equation*}  \boxed{\int_{0}^{\pi/2} \sin^{2k+1}(\theta) dx = \frac{(2k)!!}{(2k+1)!!}} \end{equation*}

This is easy to prove by induction, using integration by parts.
Click here for the proof, if you really, really want it. But you probably don't.

Seriously, no feelings will be hurt if you skip this.

Define

    \[I_{2k+1} = \int_{0}^{\pi/2} \sin^{2k+1}(\theta) d\theta\]

Then,

    \begin{equation*} \begin{split} I_{2k+1} & =  -\sin^{2k}\theta \cos\theta \bigg\rvert_{0}^{\pi/2} -\int_{0}^{\pi/2} (-\cos^2\theta) (2k) \sin^{2k-1}\theta d\theta \\ & =  (0) + (2k) \int_{0}^{\pi/2} (1 - \sin^2(\theta) \sin^{2k-1}(\theta) d\theta \\ & \Rightarrow I_{2k+1} = 2k I_{2k-1} - 2k I_{2k+1} \\ & \Rightarrow I_{2k+1} = \frac{2k}{2k+1} I_{2k-1} \\ & \Rightarrow I_{2k+1} = \frac{(2k)!!}{(2k+1)!!} \\ \end{split} \end{equation*}

Hope you feel it was worth it!

Step 3:
Start with the Taylor series for \arcsin(x):

    \[\arcsin(x) = x + \frac{1}{2} \frac{x^3}{3} + \frac{1\cdot 3}{2\cdot 4} \frac{x^5}{5} + \ldots = \sum_{k=0}^{\infty} \frac{(2k-1)!!}{(2k)!!} \frac{x^{2k+1}}{2k+1}\]

Note that this is absolutely convergent. Now simply substitute x = \sin \theta.

(3)   \begin{equation*}  \boxed{\theta = \sum_{k=0}^{\infty} \frac{(2k-1)!!}{(2k)!!} \frac{\sin(\theta)^{2k+1}}{2k+1}} \end{equation*}

Step 4:
In equation (3), integrate from 0 to \pi/2.

    \begin{equation*} \begin{split} LHS & = \int_{0}^{\pi/2} \theta d\theta \\     & = \pi^2/8  \\ RHS & = \sum_{0}^{\infty} \frac{(2k-1)!!}{(2k)!!} \frac{I_{2k+1}}{2k+1} \\     & = \sum_{0}^{\infty} \frac{(2k-1)!!}{(2k)!!} \frac{1}{2k+1} \frac{(2k)!!}{(2k+1)!!} \\     & = \sum_{0}^{\infty} \frac{1}{(2k+1)^2} \end{split} \end{equation*}

Step 5:
So the RHS is what we looked at in equation (1). This means

    \[\frac{3S}{4} = \pi^2/8\]

so

(4)   \begin{equation*} \boxed{\sum_{1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}} \end{equation*}

Addendum: I did some more research. Euler himself did something very similar (here). The earliest version of this exact proof that I could find was by Boo Rim Choe in the American Mathematical Monthly in 1987 (Vol. 94, No. 7 (Aug. – Sep., 1987), pp. 662-663 ).

Reunion

Written after my 25th college reunion.

Reunion

You’re back in your hostel — in the lounge, of course, watching tv, of course (Chhaya Geet? A Test match?). You’re sitting on one of those wicker chairs — remember how the mosquitoes could creep up from behind and bite you in the butt? Shift your gaze a little to the right from the television, along the shelves. Do you see that locked little drawer? Keep your focus on that for a little while, please.

One year, we realized that that locked drawer was, well, locked. And that it had been locked ever since anyone could remember. We broke the lock, and out of the vault came: a tape recorder and radio boombox. All reunion long, this memory kept annoying me — itching like a mosquito bite on the butt. And it took a little while, a little scratching, to figure out why this memory was so persistent.

Who were these people who left it in there, locked it up, and walked away? What were they thinking? We don’t know anything, except that they did indeed do it and then vanished. This was their Stonehenge, their Easter Island.

What about the kids in the hostels today? What if we had done that, what would they be thinking when they saw the artefact? Actually, that’s an easy question to answer. They’d be saying, what the fuck is is a “tape recorder”, because I’ve spent the last three hours trying to get it to talk to the wifi.

We are the ghosts who walk their corridors. But it’s more than that. For too long, we’ve been ghosts to each other too. Not in touch; or even if we have, we’ve been nothing more than the tiny tug on an electron in a web server in Lithuania, sending a pulse to a monitor a thousand kilometres away. We’re more than that, we’re flesh and blood, perhaps flesh and blood and a little bit of alcohol for those few days. The reunion is our reminder of this.

But it’s more than just that, too. As we see it reflected in our friends’ eyes, it is a reminder to ourselves most of all of our own existence as full human beings. We forget that in our routine; and we don’t even know we forget it. We need our own reminders.

But what of the kids we met, today’s kids, who only knew us as their ghosts? What do they think of us now they’ve seen us in person? That’s another easy answer. The bastards think we’re old; and they’re right. Wallace Stevens said that death is the mother of beauty. Nothing would matter if not for the prospect of its loss. The reminder of our flesh and blood reality is our opportunity to look around with sharper eyes at the wonder around us.

Ten of my favorite books

This was going around Facebook a while ago — what are your ten favorite books?

I’m going to restrict myself to fiction, and I’m going with the ones that affected me the most at the time I read them, rather than necessarily the best.

1) Satanic Verses, Salman Rushdie. Mindboggling to me at the time. Still wonderful. Bonus: Haroun and the Sea of Stories, by the same author, one of the best kid’s books around, both for kids and grownups.

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A manifesto for macro

I once applied (unsuccessfully) for a solo photo exhibition at a local art gallery.   They asked for a writeup to describe it; this writeup ended up being a manifesto for macro photography, why I enjoy it so much.

And here’s a link to the pdf: https://edgeofthecircle.net/macro_writeup.pdf

[pdf-embedder url=”https://edgeofthecircle.net/wp-content/uploads/2018/02/macro_writeup.pdf”]

Some thoughts on consciousness and evolution

Consciousness is a by-product of evolution.  If we try to understand what drove evolution to create consciousness, we understand the attribute itself much better.  Here are some of my notes in trying to weave together a perspective on evolution, and what light it sheds on consciousness.

Notes on Consciousness through evolution

 

An introduction to the rules of cricket

Here’s a secret: cricket’s a simple game. It’s about as complicated as baseball, and an order of magnitude less intricate than American Football. I’m not going to go into too much detail here, but here’s the truth: you don’t need to know too many rules to enjoy the sport.

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A motivation for cricket

Opinions on cricket are not hard to come by. That it’s a torturously complicated sport with rules about rules about rules, just for the pleasure of the ritual. Or it’s terribly boring, nothing ever happens and there’s nothing to watch. And the crowd consists of a bunch of old men who natter on about the iniquities of the estate taxes while waxing their moustaches.

This article is for those who want to learn more about cricket. More than that: it’s for those who are curious if there are any good reasons why they should want to learn more about cricket. I will use baseball as a reference point for some of the rules, but I hope it will be a useful read even if you don’t know any baseball.

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