How is moving one place to the left during addition of an infinite series allowed?

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In this video, the mathematician at 3:15 moves the second S₂ one place to the right, while solving for 2S₂. How is this allowed?

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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.