my problem with infinite divergent series

[phillip1882]

according to euler, the infinite series 1+2+3+4+5.... = -1/12.

the "proof" is a bit complex, but basically revovles around letting the series 1-1+1-1+1-1 = 1/2

here's my problem with this.

let S = 1 +1/2 +1/3 +1/4 +1/5 +1/6 ...

multiply S by 2.

2*S = 2 +1 +2/3 +1/2 ++2/5 +1/3 ...

subtract S

2*S -S = 2 + 2/3 +2/5 +2/7 +2/9 ...

subtract S again

2*S -S -S = 1 +2/3 -1/2 +2/5 -1/3 +2/7 -1/4 +2/9 -1/5

0 = 1 +1/6 +1/15 +1//28 +1/45... which aproaches a finite value greater than 0.

[studiot]

 

according to euler, the infinite series 1+2+3+4+5.... = -1/12.

 

 

Perhaps you would like to rephrase this?

[mathematic]

according to euler, the infinite series 1+2+3+4+5.... = -1/12.

the "proof" is a bit complex, but basically revovles around letting the series 1-1+1-1+1-1 = 1/2

here's my problem with this.

let S = 1 +1/2 +1/3 +1/4 +1/5 +1/6 ...

multiply S by 2.

2*S = 2 +1 +2/3 +1/2 ++2/5 +1/3 ...

subtract S

2*S -S = 2 + 2/3 +2/5 +2/7 +2/9 ...

subtract S again

2*S -S -S = 1 +2/3 -1/2 +2/5 -1/3 +2/7 -1/4 +2/9 -1/5

0 = 1 +1/6 +1/15 +1//28 +1/45... which aproaches a finite value greater than 0.

 

In general you can manipulate divergent series to get all sorts of results.

[imatfaal]

 

 

Perhaps you would like to rephrase this?

 

I think it was Ramanujan rather than Euler (who did a lot of work here but is never shown to have written that equation) - but the equation is right, for a particular usage of the word right

 

 

according to euler, the infinite series 1+2+3+4+5.... = -1/12.

the "proof" is a bit complex, but basically revovles around letting the series 1-1+1-1+1-1 = 1/2

here's my problem with this.

let S = 1 +1/2 +1/3 +1/4 +1/5 +1/6 ...

multiply S by 2.

2*S = 2 +1 +2/3 +1/2 ++2/5 +1/3 ...

subtract S

2*S -S = 2 + 2/3 +2/5 +2/7 +2/9 ...

subtract S again

2*S -S -S = 1 +2/3 -1/2 +2/5 -1/3 +2/7 -1/4 +2/9 -1/5

0 = 1 +1/6 +1/15 +1//28 +1/45... which aproaches a finite value greater than 0.

 

 

In general you can manipulate divergent series to get all sorts of results.

This.

[timo]

I think it was Ramanujan rather than Euler (who did a lot of work here but is never shown to have written that equation) - but the equation is right, for a particular usage of the word right

You mean when used as "obviously wrong"? I fail to see how adding up all natural numbers is supposed to a) end up in a finite result and b) in a fraction, and c) in a negative number. You have either not seen what probably is a typo or I am not getting the joke.

[studiot]

Hardy p342 Theorems on series.

 

1) A series of all positive terms must be convergent or diverge to infinity

 

2) A necessary and sufficient condition that S_n is convergent is that there exists a number K such that

 

u₀+u₁+u₂+u₃+.............u_n < K for all values of n

 

For any K, Let u_n = K

Then S_n > K

Hence by 2 the original series is not convergent

Hence by 1 it is divergent to infinity.

Edited November 18, 2015 by studiot

[imatfaal]

You mean when used as "obviously wrong"? I fail to see how adding up all natural numbers is supposed to a) end up in a finite result and b) in a fraction, and c) in a negative number. You have either not seen what probably is a typo or I am not getting the joke.

First off - can I re-iterate Mathematic's remark above. And add one of Hardy's at 13.17

"The formulae (13.10.11) give examples of the R [gothic-script] summability of divergent series of positive terms. We can use such equations, as did Euler and Ramanujan, to define the sums of series, such as 1-1+1..., of the more usual type; but the definitions which result have a narrow range and demand great caution in their application" -[my emphasis] He then goes on to show a quick source of internal contradiction if these sums are taken beyond applicability.

 

 

Hardy p342 Theorems on series.

...

 

It also states on page 333

13.10.11 that incidentally it is possible to show that 1+2+3...=-1/12

I have neither the knowledge nor the wit to explain Ramanujan summation to members - it is something "I know of" rather than something "I know about" if you can see the distinction. Here is the link to a skulls in the stars blog that I think was my first brush with the idea

 

http://skullsinthestars.com/2010/05/25/infinite-series-are-weird-redux/

 

I understand Ramanujan's heuristic but regarding the formal derivations I would have to bow out - there are loads of explanations on the web but my understanding is such that I would be weary of even recommending one over the other

[studiot]

Unfortunately p333 of my 10th edition Hardy contains miscellaneous examples on chapter VII (additional theorems in the calculus), not your referenced comment.

 

Secondly I can't trace your other references of the form 13.17 and 13.10.11.

 

However in Chapter 1V (Limits of functions of a positive integer variable) where he proves the theorems I referred to he discusses the use of sums and products of infinite series and shows that the conditions of allowability of manipulation by mathematic is that the series is convergent.

 

But (my comments now) the theorems above show that the series is not convergent so those manipulations are disallowed.

[imatfaal]

G. H. Hardy "Divergent Series"

Oxford University Press, London 1949

Reprinted - 1973

[DrP]

QUOTE from link in post 7: "... =-1/12 is incorrect as it confuses Dirichtlet series with the zeta function"

 

 

I don't quite follow those derivations.... but without following it through or spending more time than I have free to go through it all now for little reward I would assume that the quote above from the link sounds pretty reasonable.. someone has confused something somewhere if they get this result surely. I doubt anyone will actually claim that 1+2+3+4.... = -1/12 in actuality.

[studiot]

Perhaps I should have given the full reference

 

G H Hardy

 

A Course in Pure Mathematics (10 Ed 1952)

 

Cambridge.

 

I do not have his series book.

 

However the following extract from

Fort 1930

Infinite Series

Oxford

 

proves [math]\sum\limits_0^\infty {{a_a}} [/math] the infinite sum tends to infinite (Theorem 32) and the finite sum converges (Theorem 33)

 

 

 

Ferrar

 

A textbook of Convergence

 

Oxford 1959

 

Has a similar discussion to my Hardy above.

Edited November 19, 2015 by studiot

[imatfaal]

QUOTE from link in post 7: "... =-1/12 is incorrect as it confuses Dirichtlet series with the zeta function"

 

 

I don't quite follow those derivations.... but without following it through or spending more time than I have free to go through it all now for little reward I would assume that the quote above from the link sounds pretty reasonable.. someone has confused something somewhere if they get this result surely. I doubt anyone will actually claim that 1+2+3+4.... = -1/12 in actuality.

 

 

 

Not sure there are many people in history who could reasonably claim that both Euler and Ramanujan are confused. No one claims it is a universally applicable result nor comparable to the summation of a normal series nor that the concept doesn't produce contradictions - but the reality is that these counter-intuitive results are not only repeatable through different methods of analysis in some cases they explain/reproduce physical situations that otherwise are a mystery