Conclusion
This brings us into a position to answer our original question with this, the most unexpected of results. Substitute the above result into (2) .
1
1
=
=
𝜋 2 6
1 𝑘 2
∞ =1
∑
(
)
𝝅
=
= . …
Finally, we have our answer. The seemingly disordered act of picking two random integers is governed by the same universal constant that governs our understanding of circles. We can see on the course of this proof that even our most basic of numerical systems is tied to the most fundamental concepts that govern the geometry of the universe.
Bibliography
‘Basel problem’ (11/04/2017) Wikipedia . Available at: https://en.wikipedia.org/wiki/Basel_problem [Accessed: 15/04/17]
‘Maclaurin Expansion of sin(x)’ The Infinite Series Module : University of British Colombia. Available at: http://blogs.ubc.ca/infiniteseriesmodule/units/unit-3-power-series/taylor-series/maclaurin- expansion-of-sinx/ [Accessed: 15/04/17]
Sangwin, C.J. (1/12/01) An Infinite Series of Surprises . Available at: https://plus.maths.org/content/infinite-series-surprises [Accessed: 15/04/17]
‘Generating π from 1,000 random numbers’ (13/03/17) YouTube : Standupmaths. Available at: https://www.youtube.com/watch?v=RZBhSi_PwHU&t=675s [Accessed: 15/04/17]
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