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In this video, the mathematician at 3:15 moves the second Sā‚‚ one place to the right, while solving for 2Sā‚‚. How is this allowed?

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    Please take your time an post the whole question as text and not a link to a video. You may supply the link as additional information. http://math.stackexchange.com/tour ā€“ 2017-01-30

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I asked the question in the comments section of the video, and someone replied with a very simple answer:

Commutative and associative properties of addition applied to infinite sums.

So 1+2+3+4+5... = 0+1+2+3+4+5...

Never thought of it that way.

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    Careful! Commutativity is not preserved in infinite sums in general. Also, in my opinion this video is extremely misleading: they fail to specify their definition of the sum of an infinite series. Obviously all of the manipulations they do are completely nonsensical if you use the ordinary definition of limit of a sum, so that's a much bigger problem than their abuse of algebraic rules for finite sums. ā€“ 2017-03-10