Prove that 1+2+3.....infinity=-1/12?

Ah, the classical Riemann's Zeta Function. Unfortunately, that is not true, and we are only assuming the function is continuous across the y-axis (complex plane) 

Rather than prove a simple point on the Zeta Function, why not create the analytic continuation of the function like what 3b1b mentioned in his video? 

Can't be done. The "proof" found on YouTube makes the mathematical error of treating divergent series as if they were convergent and arrives at the absurd result that the divergent infinite series of positive integers converges to a negative fraction.

The errors can be described as treating the expression (∞ - ∞) as if it were a deterministic number. Divergent series cannot be manipulated to yield a deterministic result any more that 0/0 yields a deterministic result.

See the source link for a more rigorous debunking of this nonsense.