More about Euler’s crazy sums

Recently we talked about how Euler managed, through some magic tricks, to find that the sum of the inverse of the square numbers is \pi^2/6… That was a piece of virtuosismo, but even geniuses sometimes slip up. This one is funny. You may know the sum of a geometrical series:

1+x+x^2+x^3+\cdots = {1\over 1-x}

(There are many ways to understand that formula, I’ll give you a nice one soon). Now, using the same formula, sum the inverse powers:

1+x^{-1}+x^{-2}+x^{-3}+\cdots = {1\over 1-x^{-1}} = {x\over x-1}

So far, so good. Now, imagine that we want the sum of all powers, positive and negative:

\cdots+x^{-3}+x^{-2}+x^{-1}+x^0+x^1+x^2+x^3+\cdots = {1\over 1-x} + {x\over x-1} -1 = 0

Amazing!!! The sum of all powers is zero!! Nice, ein? Euler concluded that this proved the possibility of the creation of the world from nothing, ex nihilo. So, this is easy… what is the problem with this proof?

BTW, I read this story and many others in William Dunham’s book The master of us all

I'm better than you all even when I'm wrong!


4 thoughts on “More about Euler’s crazy sums

  1. The summation you used is only valid for -1<x<1. You cannot simultaneously have -1<x<1 and -1<1/x<1 fit this criteria so the summation is not valid.

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