>>11
Yes.
For instance, you can prove by induction that 1+1/2+1/4+...+1/2^n is less than 2 for any n, but the INFINITE sum 1+1/2+1/4+... is NOT less than 2.
>>12
1+1+1+1+1+1...+1 (n "1"s) is an integer for any n, but 1+1+1+1.... is infinity, which is not an integer.
Another, possibly less stupid, example: The rationals are closed under addition, so any partial sum of 1-1/3+1/5-1/7... is a rational number, but the series sums to pi/4, which isn't rational.
>>14
>it just really obviously doesn't
you fail @ math.
>>16
Ramanujan did a lot of weird stuff that turned out to be correct, but that he didn't really justify properly (and a lot of stuff that turned out to be just wrong).
This isn't the zeta function, though (because of the signs). From the zeta function you get 1+2+3+4+... = -1/12 since the sum is equal to the series
\zeta(s)=\Sigma 1/n^s evaluated at -1. Although the series converges only for Re(s) >= 1, s != 1, the analytic continuation of the function defined by this series is holomorphic for any s != 1, and it happens that
\zeta(-1)=-1/12.