proof that 1/2 = -1/2

Question

Part One:

S = 1+1+1+1+1+...

1/2 = 1-1+1-1+1-1+... *if you don't understand, google grandi's series

S - 1/2 = 2+2+2+...

S - 1/2 = 2S

S = -1/2

Part Two:

S = 1+1+1+1+...

1/2 = 1-1+1-1+1...

S + 1/2 = 2+2+2+...

S + 1/2 = 2S

S = 1/2

Conclusions:

1/2 = -1/2

Can you find the mistake?

Answer 1

The short answer is that the series $$S = 1 + 1 + 1 + ... $$ and $$G = 1 - 1 + 1-1+...$$ are divergent, and hence you cannot apply normal algebraic manipulations on them. For example Grandi's series $G$ does not equal $1/2$ in the normal sense, but through something called Cesàro summation.