The Problem With Infinity

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

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