Some years ago a spoof ( I hope) YouTube clip gave a “proof” that the sum of all the counting numbers equals -1/12, i.e. \[ \sum_{n=1}^\infty n = – \frac{1}{12} \]

Follow this link to see the video originally posted on “Numberphile”. Apparently this got several million views and many “likes” meaning many people either believed it or thought it was a good joke.

The same authors then followed this up with a second video where they referenced the Riemann zeta function as further “proof”. Video .

Some of my family saw these and wanted explanations. It was no use just saying that the Numberphile reasoning was wrong because they were rearranging terms in non convergent series – that was considered meaningless jargon. So I wrote the attached, rather dry, explanation to show the errors in the Numberphile reasoning. Each time I worked from finite series and showed what happens when we allow the number of terms to become infinite. I believe this very clearly shows the errors in the Numberphile videos.

Since then I saw another video, this time by Burkhard Polster on his site “Mathologer” which also debunks the Numberphile videos but also explains what he thinks they were trying, but failing, to state. He does this in terms of a concept called supersums, interesting viewing.