Was Pi Designed?

I don’t mean the food item, I mean the number, also known as Archimedes’s constant and the circle constant: 3.141592653589793…

In “The Artist’s Signature”, the last chapter of Carl Sagan’s novel Contact, we find that the digits of this number have a message contained in them, a message from the designer of our Universe. But such a message would only be possible if pi could have some other value. But pi has the value it does out of logical necessity, and thus could not have been designed. There are many ways to compute this number, so for illustration, I will show only one of them, 4*arctan(1):

4*(1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + …)

This is an infinite sum, but it has only one possible value. That can be shown because the partial sums of this sum form a “Cauchy sequence”, a sequence whose convergence can be deduced from the sequence members without using the series limit.

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