The things that don’t converge anywhere totally blow my mind!

In the video below, Simon is writing a sequence for pi (π), in which the sum of all the terms results in pi. Unfortunately, he got the formula that he used to calculate the terms in the sequence wrong. In Simon’s case ((-1) times (k+1)), all the terms in the sequence came out negative, while positive and negative terms alternate when using the correct formula ((-1) to the power of (k+1)). Simon writes:

Here are the actual formulas for pi:

The one written in the video:

pi/4 =  pi/4=sum_(k=1)^(infty)((-1)^(k+1))/(2k-1)

(I failed to put the equal sign next to the divide sign)

The one with the Riemann Zeta function:

 pi=sum_(k=1)^infty(3^k-1)/(4^k)zeta(k+1),

And yet it is not the calculations that made me share this video. What I love about it is that it depicts the extent to which Simon is mesmerized with math. In the second half of the video, he talks about how some complex formulas “that don’t converge anywhere” (like this pi sequence or the Riemann zeta function adding up natural numbers and resulting in negative 1/12) “totally blew my mind when I first saw this!”

The pi formulas come from here: http://mathworld.wolfram.com/PiFormulas.html

The 3Blue1Brown video visualising the Riemann zeta function:

 

Simon was also impressed by the fact that many names are likely to be found within the first billion digits of pi, if written in numbers. Here is how it works, he explained: For every word/ name, if you transform each letter into its number in the alphabet the word/name will turn into a string of numbers. If that string of numbers is under 7 digits long it is likely to be found within the first billion digits of pi! Simon’s name is too long, but his sister’s name (Neva) is most probably part of the first billion digits of pi:

DSC_3560

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