
An Elementary Proof of Wallis’ Product Formula for Pi (2005) - johnaston
http://fermatslibrary.com/s/an-elementary-proof-of-wallis-product-formula-for-pi
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mrcactu5
In order to appreciate how phenomenally basic this derivation is: here is a
derivation using Euler's infinite product expansion of the sin(x) function.

[http://www.jstor.org.sci-
hub.cc/stable/10.4169/amer.math.mon...](http://www.jstor.org.sci-
hub.cc/stable/10.4169/amer.math.monthly.119.06.518)

Then if you want your answer plug in x = π/2 inevitably Wallis formula leads
to either Stirling formula or the derivation of ζ(2)=π^2/6.

So for someone to solve it just by drawing some rectangles and finding the
area is pretty amazing.

~~~
Chinjut
One can actually derive the Euler product for sine perfectly rigorously quite
quickly, in just the manner Euler did!

(Euler is often said to have glibly moved from "sin(x)/x has roots at nonzero
integer multiples of π" to "sin(x)/x = the product of (1 - x/(kπ)) for each
nonzero integer k" in solving the Basel problem, simply ignoring the problem
of the fact that many other functions have precisely the same zeros.

But Euler was not, in fact, always so glib in presenting this argument! For
example, in Volume 1, Chapter 9 of his Introductio in Analysin Infinitorum
(translation by Ian Bruce available at
[http://www.17centurymaths.com/contents/introductiontoanalysi...](http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm)),
we see that Euler argues for the product formula by noting (what would in
modern notation be) that sin(x) = the imaginary part of e^(ix) = the imaginary
part of (1 + ix/N)^N for infinitely large N, so that a factorization of sin(x)
can be extracted from a factorization of the polynomial Im[(1 + ix/N)^N] for
large N.

I'll write out in modern style the extraction of that factorization, but all
the ideas are already present in the Euler:)

Let us denote Im[(1 + ix/N)^N] by f_N(x). Note that f_N is a polynomial of
degree either N - 1 (if N is even) or N (if N is odd), whose degree 1 term is
1. Furthermore, its roots occur where x/N is the tangent of a multiple of π/N.
Putting these together, we get that f_N(x)/x = the product of 1 - x/(N
tan(kπ/N)) over the nonzero integers k in (-N/2, N/2).

As N approaches infinity, N tan(kπ/N) approaches kπ. Thus, we have that
sin(x)/x = the product of 1 - x/(kπ) over the nonzero integers k.

[This last step is slightly glib, in that we've commuted limits without
justification. We can rectify that by bundling together the factors where k
differs only in sign, saying f_N(x)/x = the product of 1 - (x/(N tan(kπ/N)))^2
over k in (0, N/2). Now we note that the movement of the factors toward their
limit is monotonic in N (considering the k-th factor to be 1 when N/2 <= k),
which is sufficient justification for commuting the limits.]

------
ythl
I had never heard of this product formula before, so I whipped up a quick
python program: [http://pastebin.com/FBR8rWxv](http://pastebin.com/FBR8rWxv)

All I can say is wow, that formula slow to converge on pi. The product after
100 million terms is: 3.141592637878503

You can see that 100M terms only gets you 8 decimals of precision.

~~~
Twirrim
maybe I'm missing something (haven't tried to debug it), but that just results
in 0.

edit: Oh that's interesting. Python 2 gave me 0, Python 3 worked fine.

~~~
pvg
Probably missing that python 2 defaults to integer division.

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formula1
I appreciate that this is likely trivial for mathematicians. For me, I was
quite lost. Something I found a bit more obvious was constantly dividing a
squares sides into two. When done infinite amount of times, getting the
perimeter of the shape would equal 2xpixr

~~~
masterjack
I don't think anyone would call it trivial; it might be that the submitter
erroneously translates elementary as trivial. Here elementary means that it
doesn't use higher math (calculus, analysis, etc), it doesn't say anything
about ease or complication, and indeed the elementary proof of the prime
number theorem is hella complicated.

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topogios
A classic, and clear, example of "elementary" meaning something different than
"trivial" in math, is Atle Selberg's Annals proof of the Prime Number Theorem.
A link to notes on this proof, and some (!?) controversy relating to it, here:
[https://people.math.osu.edu/nevai.1/AT/ERDOS/ErdosSelbergDis...](https://people.math.osu.edu/nevai.1/AT/ERDOS/ErdosSelbergDispute.pdf)

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paulpauper
I would like to see an attempt at an 'elementary' proof of the Ramanujan pi
formulas

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jmount
As others are pointing out: the paper's title is "elementary" (not "trivial" a
in the current link title). Both of these have different and accepted meanings
in mathematics.

"Elementary" roughly means doesn't bring in results from different fields
(such as using calculus to prove a theorem about a sum or product).
"Elementary" proofs can be quite difficult, and it can be quite a feat (well
worth a publication) to find one for a standard result.

"Trivial" usually means something closer to easy, or means there is some
linkage already implying the result (just in disguise).

~~~
woopwoop
In fact, there is a negative correlation between a proof being elementary and
a proof being trivial. The reason people like abstraction is that it makes
difficult proofs trivial. The reason people don't like abstraction is that it
makes elementary proofs non-elementary.

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mercurio
Please fix the title change. 'Trivial' and 'elementary' have specific meaning
in mathematics and are not interchangeable. Roughly speaking, 'trivial' means
easy or obvious, while 'elementary' means a proof that does not use complex
analysis or higher techniques. This proof is elementary but definitely not
trivial.

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robinhouston
I don’t like the title change. The proof is elementary but non-trivial.

