November 2008


Let’s have some fun with divergent geometric series. It can be shown (or rather, argued) that 1+2+4+8+\cdots{}=-1

In fact, Euler already knew this. Intuitively one can simply try the well-known closed form for geometric series, that is

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

for x>1 (at your own risk!), to find the solution:

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

Alternatively, you can use Riemann’s Zeta function

\zeta(s) = \sum_{n=1}^{\infty} {1\over n^s}

which, with s=-2, results after heavy rewriting in -1 again.

For a different perspective, see Bill Gosper in HAKMEM 154, where he figures:

By this strategy, consider the universe, or, more precisely, algebra:

let X = \text{the sum of many powers of two} = \cdots{}111111

now add X to itself; X + X = \cdots{}111110

thus, 2X = X - 1 so X = -1

therefore algebra is run on a machine (the universe) which is twos-complement.

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