There have been a few times in my life where mathematics really puzzled me. Here is one of them.

The mathematical model in the title is mind-numbing. It suggests that continuously adding successive numbers from one to infinity would create a negative fraction.This model, however, can be explained with only three separate equations, which we'll call S(1), S(2), and S(3).

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

(This equation in itself is perplexing because it you're adding whole numbers and end up with a fraction)

Proof: S(1)=1-1+1-1+1-1+1........... ---> 1-S(1)= 1-(1-1+1-1+1-1.....) ---> distribute negative sign--> 1-S(1)=1-1+1-1+1..... ---> basically you ended up where you started---> 1-S(1)= S(1) --> solve for S(1) ---> 1=2S(1) --> S(1)=(1/2)

2. S(2)=1-2+3-4+5-6+7.........

2S(2)=1-2+3-4+5-6+7.........

+1-2+3-4+5-6..........

add together

2S(2)=1-1+1-1+1-1+1....... --> 2S(2)= S(1); S(1)=1/2, therefore 2S(2)=1/2 --> solve for S(2)

2S(2)=1/2--> S(2)=1/4

3. S(3)=1+2+3+4+5+6........ (this is the equation we are looking to solve.) --> subtract S(3)-S(2):

1+2+3+4+5+6+7........

-(1-2+3-4+5-6+ 7.......)--> (1-1)+(2+2)+(3-3)+(4+4).......--> 0+4+0+8+0+12+....... --> 4+8+12......= S(3)-S(2) 4(1+2+3+4+5....)=S(3)-S(2) -->S(2)=1/4; 4S(3)=S(3)-(1/4); solve for S(3) --> 3S(3)=-(1/4)-> Divide 3 to isolate S(3)--> S(3)=-1/12

Although this model seems counter-intuitive, this result appears several times in concepts such as string theory.