Without differentiation or integration, or number theory, could we still prove this infinite sum? | Visit http://brilliant.org/Mathemaniac/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.

π/4 = 1 - 1/3 + 1/5 -..., also called the Leibniz series, is quite famous, but the usual proof involves differentiation or integration. The more visual geometric proof still relies a lot on some advanced theorems from number theory, but given how simple the series is, is it possible to have an even simpler proof? Yes! And this video tries to explain this.

Actually, similar to the previous proof, this proof has been at the video ideas list for quite a long time - so it's good to finally see this out!

By the way, on second thought, this looks similar to Fourier coefficients, at least the time average bit - though I can't see whether this is the same proof as the one using Fourier series of sgn(x). It feels very connected, but also very different in the sense that we don't need to use the Cesàro sums of the Fourier series. Please let me know if you have any ideas regarding this.

Video chapters:
00:00 Intro
00:45 Chapter 1: The setup
03:16 Chapter 2: The main argument
14:55 Chapter 3: Making this rigorous
17:33 Sponsored segment

This video was sponsored by Brilliant.

Sources (in order of appearance in the video):

(1) Pi hiding in prime regularities (3Blue1Brown):    • Pi hiding in prime regularities  

(2) Fermat’s Christmas Theorem (Mathologer):    • Fermat's Christmas theorem: Visualisi...  

(3) The proof itself: https://www.quora.com/How-can-I-show-...

(4) Cesàro sums: https://en.wikipedia.org/wiki/Ces%C3%...

A little more reading:

(1) Calculating pi by hand (Matt Parker):    • Calculating π by hand  

(2) Regular sum, if defined, equals Cesàro sum: https://math.stackexchange.com/questi...
https://math.stackexchange.com/questi...

(3) Wikipedia proof (involving integration): https://en.wikipedia.org/wiki/Leibniz...
(Worth noting that this requires a bit more than just plugging in x = 1 in arctangent series - this requires the justification from Abel’s theorem)

(4) If you want to prove the Cesàro sum of the positions to be 0 more rigorously, here is the justification:
https://www.wolframalpha.com/input?i=...
and
https://www.wolframalpha.com/input?i=...
(and the formula in the first link would only require repeatedly applying product to sum formulas and telescoping series - still no differentiation / integration involved. If you want even further assistance, I made a video on this:    • Algebra vs Geometry in Trigonometry |...   )

Music:
Asher Fulero - Beseeched
Aakash Gandhi - Heavenly, Kiss the Sky, Lifting Dreams
(From YouTube audio library)

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