## 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 \[n=1,2,3,...\]