1 + 2 + 3 + 4 + ... = - 1/12

Let's make it clear: 1+2+3+... is a divergent series, admitting infinity as a limit.

However, there are multiple ways of defining a "summation" of the series. Ramanujan's technique confirms the analytic continuation of Riemann's zeta function:

  zeta(-1) -> -1/12

So the equal sign actually means "there exists a summation technique leading to that result". That's why the original formula reads:

  1 + 2 + 3 + 4 ... = -1/12 (R)

The (R) indicates Ramanujan's specific summation.

This has already been posted multiple times in the past few days : https://news.ycombinator.com/item?id=7057049 https://www.hnsearch.com/search#request/all&q=1%2F12

-1/12 (R), misleading title

this infinite sum was featured in a numberphile video a few days ago with a entraining and simple "proof": http://www.youtube.com/watch?v=w-I6XTVZXww

and a more advanced explanation & proof in the bonus-video: http://www.youtube.com/watch?v=E-d9mgo8FGk

you should definitely subscribe to numberphile (http://www.youtube.com/user/numberphile) and their sister channel computerphile (http://www.youtube.com/user/computerphile) on youtube :)

Those of us of a less mathsy persuasion may wish to watch a more layman-friendly explanation:

https://www.youtube.com/watch?v=w-I6XTVZXww