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So isn't this series basically sigma of n for which n starts from 1 and ends at infinity? In this case, in my opinion, as every terms in this series is positive, this series can immediately be included in the 'comparison tests for convergence and divergence'. French mathematician Nicole Oresme has proved the divergence of the harmonic series 1/n by proving the divergence of a series which has lesser terms than that of the harmonic series. If, as what the numberphile people are saying, 1+2+3+4+... forever does equals to 1/12 doesn't this make the whole concept of comparison test flawed? Even if the Grandi's series which is 11+11+11... does actually equals to a half, the answer 1+2+3+4+.. forever does not make sense as it makes previous proofs and statements made in infinite series incorrect.
Also, i don't understand what the average partial sums are for. Why is there the need to average the partial sums? The average of the partial sums may converge towards a half but that doesn't mean anything that the 'nonaverage' (as in real) series converge towards a half...