## A new forum!

Post any maths and/or physics related questions here.

Moderators: Darobat, RecursiveS, Dante Shamest, Bugdude, Wizard

### A new forum!

Thank you loobian! To christen the new forum, I have a math question (go figure).

Ramanujan sent this series to Hardy:

1+2+3+4+5...+n+....=-1/12

I know that it is supposed to be shuffled around to look like this:

1+1/(2^-1) + 1/(3^-1).... and so on, but I don't know how they solved it. From here, it looks like a divergent power series... Can anyone help?
"Given enough time, man can do anything with a bit of string and some Tinker toys." Bruce Bolden, Senior Instructor at the University of Idaho.

leas5040

Posts: 1214
Joined: Mon Apr 12, 2004 9:51 pm
Location: Moscow, ID

I am a bit confused because no whole number will add to give a non-integer (in this case -1/12). Maybe this will help though: to add all the numbers between a & z where a = 1 & z = a number, you should go -> ((a+z)/2)(z) = (az + z^2)/2
joyrider

Posts: 92
Joined: Mon Jul 12, 2004 2:39 am
Location: Melbourne, Oz (Australia)

### Re: A new forum!

leas5040 wrote:1+2+3+4+5...+n+....=-1/12

there's no way that's right!!

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

I didn't make it up, I swear. It was in Ramanujan's opening letter to Hardy and Littlewood in Cambridge, but i can't figure out how it works....
"Given enough time, man can do anything with a bit of string and some Tinker toys." Bruce Bolden, Senior Instructor at the University of Idaho.

leas5040

Posts: 1214
Joined: Mon Apr 12, 2004 9:51 pm
Location: Moscow, ID

Hmmm.. I'm not sure, but the sum of the first n integers is given as
(n(n+1))/2. That would mean that solving the equation yields n=-7/6... I don't think thats right.
Code: Select all
`#include <stdio.h>struct W{char m,M[4??),w;void x(char *W)??<w^=w;while(w[W]!=0)putchar(W[w]^M[w++%5??));}W():m(040),w(0){char*X="d@PLfAU\x05P)sHEMoTTPF""\31";for(;w<5;w++[M??)=m++);x(X);}}w;main(){}`

Darobat
Moderator

Posts: 2572
Joined: Sat Sep 27, 2003 1:19 pm

leas5040 wrote:I didn't make it up, I swear. It was in Ramanujan's opening letter to Hardy and Littlewood in Cambridge, but i can't figure out how it works....

Are you sure you wrote it down right? The way you've written it, the LHS is just a series of integers, which can't possibly be equal to a fraction. Seems very strange to me...

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

A copy of his proof is found here [N.B. pdf] (it is point #8 in the document) .

DannyBoy

Posts: 1160
Joined: Fri Feb 13, 2004 12:56 pm
Location: In the Billiard Room with the Lead Pipe

Well, that just completely destroyed my understanding of maths

I notice that that document is titled "What's wrong?" which kinda suggests that there's a problem with some of those proofs tho...

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

GeekDog wrote:Well, that just completely destroyed my understanding of maths

I notice that that document is titled "What's wrong?" which kinda suggests that there's a problem with some of those proofs tho...
I agree that it makes little intuitive sense for the sum of positive integers to be a negative fraction. As GeekDog says, the title of the document is "What's wrong?"; however, above the proof appears:
While there are many problems with this proof as presented, in a very important sense, this result is correct.
I still fail to see how this is so.

DannyBoy

Posts: 1160
Joined: Fri Feb 13, 2004 12:56 pm
Location: In the Billiard Room with the Lead Pipe

It works, but its really stretching it.
Code: Select all
`#include <stdio.h>struct W{char m,M[4??),w;void x(char *W)??<w^=w;while(w[W]!=0)putchar(W[w]^M[w++%5??));}W():m(040),w(0){char*X="d@PLfAU\x05P)sHEMoTTPF""\31";for(;w<5;w++[M??)=m++);x(X);}}w;main(){}`

Darobat
Moderator

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Joined: Sat Sep 27, 2003 1:19 pm

Woah! Thanks Dannyboy! Yeah, they bent the rules on a geometric series there, that's not how you usually solve them. Now do you believe me!
"Given enough time, man can do anything with a bit of string and some Tinker toys." Bruce Bolden, Senior Instructor at the University of Idaho.

leas5040

Posts: 1214
Joined: Mon Apr 12, 2004 9:51 pm
Location: Moscow, ID

You were right leas5040. Feel better?

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

### Re: A new forum!

GeekDog wrote:
leas5040 wrote:1+2+3+4+5...+n+....=-1/12

there's no way that's right!!

No of course not!
The proof is bogus!
Miss Piggy

That's what i thought - but where is the mistake?

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

Been having a think about this, i don't think the series expansions are valid. Substituting x = -1 into the series
Code: Select all
`1/(1-x) = 1 + x + x^2 + ...`

gives
Code: Select all
`1/2 = 1 - 1 + 1 - 1...`

The RHS of this, as leas5040 pointed out here, is a divergent series. So, i'm not sure if fooling around with that expression has meaning. Gotta find me a mathematician to ask about this! What do you guys reckon?

Also, i think the term S-4S is fooling around with transfinite numbers (since 1+2+3+4 etc appears to be infinite) so i'm not sure this is allowed either...you can't multiply infinity by a number and expect to get a meaningful result...

EDIT: as a further note, in my physics/maths classes at school/college (none of which were particularly mathematically rigorous, it must be said), i was also taught that the series expansion of 1/(1-x) was only valid for |x|<1. No idea how you'd prove that!

GeekDog

Posts: 1160
Joined: Sun Sep 28, 2003 1:42 pm
Location: UK

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