add the infinite sum S=1+2+3+4+5.... infinity and you get a negative number. Hardy
OT Sum of real positive numbers to infinity
Started by ●February 12, 2014
Reply by ●February 12, 20142014-02-12
On Tue, 11 Feb 2014 23:47:10 -0800 (PST), gyansorova@gmail.com wrote:>add the infinite sum > >S=1+2+3+4+5.... infinity > >and you get a negative number. > > > >HardyYou are a victim of vigorous handwaving. Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●February 12, 20142014-02-12
On Thursday, February 13, 2014 5:51:41 AM UTC+13, Eric Jacobsen wrote:> On Tue, 11 Feb 2014 23:47:10 -0800 (PST), gyansorova@gmail.com wrote: > > > > >add the infinite sum > > > > > >S=1+2+3+4+5.... infinity > > > > > >and you get a negative number. > > > > > > > > > > > >Hardy > > > > You are a victim of vigorous handwaving. > > > > > > Eric Jacobsen > > Anchor Hill Communications > > http://www.anchorhill.comhttp://kottke.org/14/01/the-sum-of-all-positive-integers The sum of all positive integers  JAN 16 2014 What do you think you get if you add 1+2+3+4+5+... all the way on up to infinity? Probably a massively huge number, right? Nope. You get a small negative number: This is, by a wide margin, the most noodle-bending counterintuitive thing I have ever seen. Mathematician Leonard Euler actually proved this result in 1735, but the result was only made rigorous later and now physicists have been seeing this result actually show up in nature. Amazing.
Reply by ●February 12, 20142014-02-12
On 2/12/14 11:56 AM, gyansorova@gmail.com wrote:> On Thursday, February 13, 2014 5:51:41 AM UTC+13, Eric Jacobsen wrote: >> On Tue, 11 Feb 2014 23:47:10 -0800 (PST), gyansorova@gmail.com wrote: >> >> >> >>> add the infinite sum >> >>> >> >>> S=1+2+3+4+5.... infinity >> >>> >> >>> and you get a negative number. >> >>> >> >>> >> >>> >> >>> Hardy >> >> >> >> You are a victim of vigorous handwaving. >> >> >> >> >> >> Eric Jacobsen >> >> Anchor Hill Communications >> >> http://www.anchorhill.com > http://kottke.org/14/01/the-sum-of-all-positive-integers > > > > The sum of all positive integers  JAN 16 2014 > What do you think you get if you add 1+2+3+4+5+... all the way on up to infinity? Probably a massively huge number, right? Nope. You get a small negative number: > > > > This is, by a wide margin, the most noodle-bending counterintuitive thing I have ever seen. Mathematician Leonard Euler actually proved this result in 1735, but the result was only made rigorous later and now physicists have been seeing this result actually show up in nature. Amazing.i don't know how to type how much i'm giggling. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 12, 20142014-02-12
Reply by ●February 12, 20142014-02-12
On Thursday, February 13, 2014 7:47:57 AM UTC+13, julius wrote:> Nobody panic, it's a divergent series. All is well.That's the whole point, it isn't! It sums to -1/2 I think. The proof is quite straightforward,
Reply by ●February 12, 20142014-02-12
On Wed, 12 Feb 2014 08:56:41 -0800, gyansorova wrote:> On Thursday, February 13, 2014 5:51:41 AM UTC+13, Eric Jacobsen wrote: >> On Tue, 11 Feb 2014 23:47:10 -0800 (PST), gyansorova@gmail.com wrote: >> >> >> >> >add the infinite sum >> >> >> > >> >S=1+2+3+4+5.... infinity >> >> >> > >> >and you get a negative number. >> >> >> > >> >> > >> >> > >> >Hardy >> >> >> >> You are a victim of vigorous handwaving. >> >> >> >> >> >> Eric Jacobsen >> >> Anchor Hill Communications >> >> http://www.anchorhill.com > http://kottke.org/14/01/the-sum-of-all-positive-integers > > > > The sum of all positive integers  JAN 16 2014 What do you think you get > if you add 1+2+3+4+5+... all the way on up to infinity? Probably a > massively huge number, right? Nope. You get a small negative number: > > > > This is, by a wide margin, the most noodle-bending counterintuitive > thing I have ever seen. Mathematician Leonard Euler actually proved this > result in 1735, but the result was only made rigorous later and now > physicists have been seeing this result actually show up in nature. > Amazing.The Euler proof that they point to is for a different series, and it coughs up a positive number. The Euler proof in the cited pdf looks like the kind of thing that Euler may have dragged out after he and his colleagues had finished off a bottle of Schnapps, and were looking for ways to boggle one another. The counter-proof is in the second line: 1 + 2x + 3x^2 + ... + (n+1)x^n = 1/(1 - x)^2 Take the limit of this as x approaches +1 from the left, and you get infinity. Yes, if you test this theory on a modern computer, in C using int, you'll get some negative numbers -- but that's because of 2's compliment arithmetic, not any fundamental truths. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●February 12, 20142014-02-12
On Wed, 12 Feb 2014 11:33:56 -0800, gyansorova wrote:> On Thursday, February 13, 2014 7:47:57 AM UTC+13, julius wrote: >> Nobody panic, it's a divergent series. All is well. > > That's the whole point, it isn't! It sums to -1/2 I think. The proof is > quite straightforward,The proof is quite straightforward, by the simple expedient of taking an analytical function outside of the range in which it is convergent. Which is something you can only get past your local trained mathematician if you have a bottle of Schnapps on hand. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●February 12, 20142014-02-12
On Wed, 12 Feb 2014 14:07:35 -0600, Tim Wescott <tim@seemywebsite.really> wrote:>On Wed, 12 Feb 2014 11:33:56 -0800, gyansorova wrote: > >> On Thursday, February 13, 2014 7:47:57 AM UTC+13, julius wrote: >>> Nobody panic, it's a divergent series. All is well. >> >> That's the whole point, it isn't! It sums to -1/2 I think. The proof is >> quite straightforward, > >The proof is quite straightforward, by the simple expedient of taking an >analytical function outside of the range in which it is convergent. > >Which is something you can only get past your local trained mathematician >if you have a bottle of Schnapps on hand.There are lots of application specific reasons to use analytic continuations of functions. One of which appears to be trolling new groups. Mark
Reply by ●February 12, 20142014-02-12
On Wed, 12 Feb 2014 14:40:58 -0800, Mac Decman wrote:> On Wed, 12 Feb 2014 14:07:35 -0600, Tim Wescott > <tim@seemywebsite.really> wrote: > >>On Wed, 12 Feb 2014 11:33:56 -0800, gyansorova wrote: >> >>> On Thursday, February 13, 2014 7:47:57 AM UTC+13, julius wrote: >>>> Nobody panic, it's a divergent series. All is well. >>> >>> That's the whole point, it isn't! It sums to -1/2 I think. The proof >>> is quite straightforward, >> >>The proof is quite straightforward, by the simple expedient of taking an >>analytical function outside of the range in which it is convergent. >> >>Which is something you can only get past your local trained >>mathematician if you have a bottle of Schnapps on hand. > > There are lots of application specific reasons to use analytic > continuations of functions. One of which appears to be trolling new > groups. > > Mark:) -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
