4 ms·

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

2 points by sz4kerto 12y ago

ColinWright 12y ago
This has been discussed a lot, but there is something real and genuine going on here that is, unfortunately, masked by the obvious nonsense. Briefly let me introduce the main ideas. When -1<x<1, sum_{n=0}^{n=oo} x^n = 1/(1-x). We might be tempted, then, to say that 1 + x + x^2 + x^3 + x^4 + ... = 1/(1-x) But that's obvious nonsense when, say, x=2. The LHS clearly makes no sense, even though the RHS exists and has the very sensible value of -1. It therefore cannot be said that: 1 + 2 + 4 + 8 + 16 + ... = -1. However, underlying all this is the concept of Analytic Continuation which says that under certain circumstances, this is a reasonable thing to be doing. Sort of. Here it is being done properly: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri... Here is a Hacker News discussion of that: https://news.ycombinator.com/item?id=7078744 https://news.ycombinator.com/item?id=7078744 If that has whetted your appetite for more you might like to follow some of the discussions and links given in previous submissions: https://news.ycombinator.com/item?id=7078489 https://news.ycombinator.com/item?id=7078489 : 53 comments https://news.ycombinator.com/item?id=7073976 https://news.ycombinator.com/item?id=7073976 : 29 comments https://news.ycombinator.com/item?id=7057049 https://news.ycombinator.com/item?id=7057049 : 19 comments https://news.ycombinator.com/item?id=7038809 https://news.ycombinator.com/item?id=7038809 : 6 comments https://news.ycombinator.com/item?id=7081885 https://news.ycombinator.com/item?id=7081885 : 4 comments https://news.ycombinator.com/item?id=8843219 https://news.ycombinator.com/item?id=8843219 : 4 comments Other submissions: https://news.ycombinator.com/item?id=7033444 https://news.ycombinator.com/item?id=7033444 https://news.ycombinator.com/item?id=7074820 https://news.ycombinator.com/item?id=7074820 https://news.ycombinator.com/item?id=7079921 https://news.ycombinator.com/item?id=7079921 https://news.ycombinator.com/item?id=7096326 https://news.ycombinator.com/item?id=7096326 https://news.ycombinator.com/item?id=7176024 https://news.ycombinator.com/item?id=7176024 https://news.ycombinator.com/item?id=7602317 https://news.ycombinator.com/item?id=7602317 https://news.ycombinator.com/item?id=7638147 https://news.ycombinator.com/item?id=7638147

2 points by sz4kerto 12y ago

This has been discussed a lot, but there is something real and genuine going on here that is, unfortunately, masked by the obvious nonsense. Briefly let me introduce the main ideas. When -1<x<1, sum_{n=0}^{n=oo} x^n = 1/(1-x). We might be tempted, then, to say that 1 + x + x^2 + x^3 + x^4 + ... = 1/(1-x) But that's obvious nonsense when, say, x=2. The LHS clearly makes no sense, even though the RHS exists and has the very sensible value of -1. It therefore cannot be said that: 1 + 2 + 4 + 8 + 16 + ... = -1. However, underlying all this is the concept of Analytic Continuation which says that under certain circumstances, this is a reasonable thing to be doing. Sort of. Here it is being done properly: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri... Here is a Hacker News discussion of that: https://news.ycombinator.com/item?id=7078744 https://news.ycombinator.com/item?id=7078744 If that has whetted your appetite for more you might like to follow some of the discussions and links given in previous submissions: https://news.ycombinator.com/item?id=7078489 https://news.ycombinator.com/item?id=7078489 : 53 comments https://news.ycombinator.com/item?id=7073976 https://news.ycombinator.com/item?id=7073976 : 29 comments https://news.ycombinator.com/item?id=7057049 https://news.ycombinator.com/item?id=7057049 : 19 comments https://news.ycombinator.com/item?id=7038809 https://news.ycombinator.com/item?id=7038809 : 6 comments https://news.ycombinator.com/item?id=7081885 https://news.ycombinator.com/item?id=7081885 : 4 comments https://news.ycombinator.com/item?id=8843219 https://news.ycombinator.com/item?id=8843219 : 4 comments Other submissions: https://news.ycombinator.com/item?id=7033444 https://news.ycombinator.com/item?id=7033444 https://news.ycombinator.com/item?id=7074820 https://news.ycombinator.com/item?id=7074820 https://news.ycombinator.com/item?id=7079921 https://news.ycombinator.com/item?id=7079921 https://news.ycombinator.com/item?id=7096326 https://news.ycombinator.com/item?id=7096326 https://news.ycombinator.com/item?id=7176024 https://news.ycombinator.com/item?id=7176024 https://news.ycombinator.com/item?id=7602317 https://news.ycombinator.com/item?id=7602317 https://news.ycombinator.com/item?id=7638147 https://news.ycombinator.com/item?id=7638147