&quot;The sum of the series 1+2+3+4+5+6... = -1&#x2F;12&quot; is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series.<p>Even mathematicians working with Cesàro sums surround such statements with &quot;this holds <i></i>only if<i></i> we interpret the infinite sum defining Z to be the Cesàro sum...&quot; [0]<p>Precisely none of the times I&#x27;ve heard the &quot;1+2+3+4...=-1&#x2F;12&quot; bullshit has the person stating it prefaced their statement with &quot;this holds <i></i>only if<i></i> we interpret the infinite sum defining Z to be the Cesàro sum...&quot;<p>If you say that &quot;1+2+3+4...=-1&#x2F;12&quot; without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that &quot;this 95 cent pack of gum is free&quot; because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum.<p>The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers.<p>[0] Madras, Neal. &quot;A Note on Diffusion State Distance.&quot; arXiv preprint arXiv:1502.07315 (2015).