>>12355396>>12355344You're a genuine idiot and the only maths PhD you could possibly have is from some shithole university where a PhD means shit.
>>12355079>>12355152You are correct. The actual sum (standard definition by the limit of partial series) is infinity (which means the partial sums get arbitrarily large).
The point is that the proofs you've seen show that if this series actually does have a real value, then following some intuitive transformations that value is -1/12. But it doesn't (under the standard summation), so the conclusion is false.
In a similar way you take a look at the series 1+2+4+.... If it does have a value, say S, then 2S = 2 + 4 + 8 + ... = S - 1, which means S=-1.
This shows if the series has a value (it doesn't) then the value must be -1.
However, there are nonstandard summations where 1+2+4+... actually does have a value. For that you need to change the meaning of convergence by introducing a new metric on the rationals Q called the dyadic metric (look it up, it's not hard to define). With this new metric, call it d, if A_n denotes 1+2+4+...+2^n, we can prove that d(A_n , -1) tends to 0 so we can say that A_n converges to -1 under the dyadic metric.
It's similar with the series 1+2+3+.... There are alternative ways to define an infinite sum which satisfy some nice properties and under which the series 1+2+3+... actually DOES have a value, which is -1/12. This is not, however, the standard summation, which is the point that confuses so many people.
