## The Problem With Infinity

Infinities don't behave like proper numbers. It makes sense to write
$x--x=0$
when
$x$
is an ordinary number but not when
$x$
is infinity. This is because when
$x$
is multiplied by 2 the result is
$2x$
but when infinity is infinity is multiplied by infinity, the result is infinity.
Because of this following proof contains an error.
$S=1++2+4+8+16+32+...+ \infty +...$
(1)
$2S=2+4+8+16+32+...+ \infty +...$
(2)
(2)-(1) gives
$S=-1$
. This is obviously wring since all terms rare positive. The above technique will only work if
$\frac{a_{n+1}}{a_n} =r<1$
for
$n=1,2,3,...$