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Some popular math videos point out that, for example, the value of

Is there an elementary way to connect this with the more formal definitions of divergent sums, e.g. analytic continuation? By elementary, I mean something that the popular math channels (e.g. Mathologer or Numberphile) might come up with, to show that the two procedures are somehow equivalent.

Also, it seems that this can be generalized to all sorts of arbitrary sums via Taylor's theorem -- is that the case?

Edit: Are there cases where this fails?

**-1/12**for the divergent sum 1 + 2 + 3 + 4 .... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on.Is there an elementary way to connect this with the more formal definitions of divergent sums, e.g. analytic continuation? By elementary, I mean something that the popular math channels (e.g. Mathologer or Numberphile) might come up with, to show that the two procedures are somehow equivalent.

Also, it seems that this can be generalized to all sorts of arbitrary sums via Taylor's theorem -- is that the case?

Edit: Are there cases where this fails?

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