An Ordinary Series
Translated from Russian. Originally published on Nov 6, 2007 as “Обычный ряд”.
Curious: take the sum
S = 1 + 2 + 4 + 8 + 32 + ...,
and multiply it by two:
2S = 2 + 4 + 8 + 32 + 64 + ... => 2S = S - 1,
and we get that our sum S equals -1.
So 1 + 2 + 4 + ... is an infinitely precise representation of -1 in a binary computer.
P.S.: Of course, this is no paradox at all — you just have to handle sums properly.