Even if you're not a maths nerd (I'm not), this is pretty bloody cool!
staaahhp trying to br8 math. dis makes me really crii.:'< by abel's definition, the sum of an infinite sum exists iff its associated sequence of partial sums converges to a limit as n->infinity.
the sequence of partial sums for s1 is {1,0,1,0,1,0,1,...} which diverges by bounded oscillation so s1 is digergent. you shouldn't just take the average of its individual terms and and say that the s1=1/2, because no matter how long you add for, you will never actually get to 1/2.
And you can't use the algebra of series with non convergent series, so he shouldn't be able to use s1 to calculate s2 and finally s. Which is why the conclusion that s=1+2+3+4+5+...=-1/12 seems like such a counter intuitive answer.
if we're going to do dodgy mathematics, i could just as easily say that s2->infinity rather than s2=1/4 by pooping out something like this.

which is complete bs because s2's sequence of partial sums diverges by unbounded oscillation. :3 and i could've manipulated it to ->-inf just as easily.
(this is why we need precise definitions in mathematics)