Ok here goes;

Make a sequence of numbers and call it ( * );

1, -1/2, 1/2, -1/3, 1/3, -1/4, 1/4, ...

Now look at the sum of this sequence. Clearly all terms except the 1 cancel out (you substract something, then add the same thing again), and the sum is 1.

Now we make the sum a bit more interesting;

(1-1/2) + (1/2-1/3) + (1/3-1/4) + (1/4-1/5) + ... = 1

We note that 1/7-1/8 = (8-7)/(8*7) = 1/(8*7), similarly for the other terms we create this formula;

S = 1/2 + 1/6 + 1/12 + 1/20 + ... = 1.

Now to make it even more interesting;

You can see that 1/(3*4) = -2/3 + 3/4, and 1/(8*9) = -7/8 + 8/9 and so on.

Now we substitute that in S;

S = 1/2 + 1/6 + 1/12 + ... = 1/2 -1/2 +2/3 -2/3 +3/4 -3/4 +4/5 -...=0

That is strange, for jsut a minute ago we saw that S=1, tohugh now we get S=0 !

It might seem that the last term doesn't have a sucessor to cancel with, but that's an illusion, for there is no last term. Every number there is followed immedeately by a negative term that cancels it out.

So the sum of 1/2 + 1/6 + 1/12 +... =0. Kinda strange since all terms are greater then 0.

Now we change the order of ( * ), we take one positive term, and then two negative ones, then a positive one, and so on. Call this ( ** );

1, -1/2, -1/3, 1/2, -1/4, -1/5, 1/3, ...

The sum of this sequence is also 1, since addition is commutative (a+b=b+a for all a,b) and the sum of ( * ) was 1.

Now we have;

1= 1+ (-1/2) + (-1/3) + 1/2 + ...=

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

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

1/6 + 1/20 + 1/42 + ... <

1/2 + 1/6 + 1/12 + 1/20 +...

So 1/2 + 1/6 + 1/12 + ... is not 1, not 0, but more then 1!

Now we change ( * ) again. Now we take one positive term, followed by 3 negative terms. Then we have (you can verify it);

1= (1-1/2-1/3-1/4) + (1/2-1/5-1/6-1/7) + (1/3-1/8-1/9-1/10)+ ... =

-2/(2*3*4) + -2/5*6*7 + -2/8*9*10 + ... < 0

So now the sum is less then 0!