A new forum!

[leas5040]

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?

[joyrider]

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

[GeekDog]

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

there's no way that's right!!

[leas5040]

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....

[Darobat]

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.

[GeekDog]

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...

[DannyBoy]

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

[GeekDog]

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...

[DannyBoy]

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.

[Darobat]

It works, but its really stretching it.

[leas5040]

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!

[GeekDog]

You were right leas5040. Feel better?

[Miss Piggy]

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

there's no way that's right!!

No of course not!
The proof is bogus!

[GeekDog]

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

[GeekDog]

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!