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Old 2020-04-02, 06:54   #17
kriesel's Avatar
Mar 2017
US midwest

107478 Posts

L'Hopital's rule for determining the value of ratio of infinities as they approach a limit applies, and allows comparison of sum of series B / sum of series A even though both sums are infinite for infinite number of elements. (I think my calculus professor would approve.) The video linked in the original post makes clear the 3-dot symbology ... represents an infinite series' unwritten further elements, when referring to the series written S=1+2+3+4+5+6 ... as all the natural numbers out to infinity.

S=-1/12 =1+2+3+4+5+6 ... does not work for me. Adding positive elements moves the sum further positive at every increment, all infinity of them. It amounts to saying that the number line is curved, intersecting itself at +inf and -1/12 (and perhaps there are other cases the men in the video could come up with too). What is there to bend a Euclidean geometry, one-dimensional, ideal, straight line back onto itself?

To have a noninteger sum of integers is a strange result.
To have a negative sum of positive integers is a strange result.
To have the absolute value of the sum smaller than the absolute value of any of the elements whose absolute values are all additive is a strange result.
To have the absolute value of the sum smaller than the least element or the least difference of the absolute values of the summed terms are strange results.
Pretty much any of these would be considered indications of error.
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