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Dear David and Marianne,
Thank you for the interesting explanations. I would remark, though, that the summation "1-1+1-1+... = 1/2" is not the source of fallacy in the Numberphile video. That series, while not convergent, is still "Cesaro summable" [which means, that the average value of its first n partial sums converges in the limit as n is taken to infinity].
The Cesaro sum of series has many nice properties, e.g., allowing a certain class of rearrangements of the terms, allowing the sums to be added when Cesaro summable series are added term by term, etc. The fallacies enter elsewhere, in the various manipulations of divergent and non-Cesaro summable series. Such manipulations can be used to arrive at any value at all for the sum of all the natural numbers.
Perhaps others have already made this point, as I've not viewed all of the linked and related sites.
Like others, I too would have wished to see some indication in the Numberphile video of the "tongue in cheek" nature of the "proof," without losing the huge entertainment value. Fallacies (like "proving" that 1 = 0 with a hidden divisioon by zero in the algebra) have always been a source of fun and exploration in school mathematics.
Here, the video is understandable to many middle grades students as well as high school students; this is great! But their teachers then need to know it is "sleight of hand," and encourage their students to question the "authority" behind it. Teaching mathematics by authority rather than through understanding is a continuing, unfortunate trend in our schools, at least in the USA, and I cannot help but feel an opportunity here might be being missed.
Gerald Goldin
Rutgers University