Some sample about infinite sum

First, let we see a infinite sum: S = 1+1/2+1/4+1/8+1/16+..... is equal to 2,beacause if we get

2S = 2+1+1/2+1/4+1/8+...=2+S => S= 2.

On the other hand,this same reasoning suggests that we ought to define T=1+2+4+8+16= -1,because

2T = 2+4+8+16+32+... = T-1 => T = -1.

Something funny is going on;how can we get a negative number by summing positive quantities?

The main reason is that S and T is different, S has the sum,but T does not.