I have seen this movie before (literally). Go down this path and discover that the sum of all natural numbers equals -1/12.

\(\zeta(-1)=-\frac{1}{12}\)

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- Cuchulainn
**Posts:**59389**Joined:****Location:**Amsterdam-
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It's Grandi's series and its Cesaro sum is 1/2.indeed, IF S EXISTS THEN1/2S = 1 - 1 + 1 -1 + 1 .... (forever)

Find S

S-1 = -1 +1 -1 +1 .... (forever) =-S

so 2S=1

It is a technique to assign values to infinite sums that are not convergent in the usual sense. Useful for Fejer's theorem and Fourier series.

(Apologies for no accent on the names; Wilmott forum does not support them?)

Cesàro, you can cut-and-paste accented letters from elsewhere

if you call [$]S_{n}[$] the sum of the first [$]n[$] terms (partial sum) then [$]S_{n}[$] is 0 if [$]n[$] is even and 1 if [$]n[$] is odd.

so [$]\lim_{n\to\infty}S_{n}[$] does not exist

if you call [$]S_{n}[$] the sum of the first [$]n[$] terms (partial sum) then [$]S_{n}[$] is 0 if [$]n[$] is even and 1 if [$]n[$] is odd.

so [$]\lim_{n\to\infty}S_{n}[$] does not exist

- katastrofa
**Posts:**7937**Joined:****Location:**Alpha Centauri

Indeed if S-1 = -S, then S = 1/2. Cauchy and Weierstrass turn in their gravesindeed, IF S EXISTS THEN1/2S = 1 - 1 + 1 -1 + 1 .... (forever)

Find S

S-1 = -1 +1 -1 +1 .... (forever) =-S

so 2S=1

the [$]n[$]th-term test for divergence:

If [$]\lim_{n\to\infty}a_{n}\ne 0[$] or if the limit does not exist, then [$]\sum _{n=1}^{\infty }a_{n}[$] diverges

Which is what we've got here

If [$]\lim_{n\to\infty}a_{n}\ne 0[$] or if the limit does not exist, then [$]\sum _{n=1}^{\infty }a_{n}[$] diverges

Which is what we've got here

- Cuchulainn
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Maybe, but Cauchy would accept it but claim it is not a Cauchy sequence. However,. he might say the Cearo sum does form a Cauchy sequence. And the bespoke sequence does not satisfy the assumptions of the Weierstraß M-test. The Cesaro partial sums form a sequence (1/2, 1/2, 2/3, 2/4, 3/5, 4/7, 4/8,...).Indeed if S-1 = -S, then S = 1/2. Cauchy and Weierstrass turn in their gravesindeed, IF S EXISTS THEN

1/2

S-1 = -1 +1 -1 +1 .... (forever) =-S

so 2S=1

Maybe they did not think about things they did not think about.

- FaridMoussaoui
**Posts:**400**Joined:****Location:**Genève, Genf, Ginevra, Geneva

This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/\(\zeta(-1)=-\frac{1}{12}\)

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- Cuchulainn
**Posts:**59389**Joined:****Location:**Amsterdam-
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(.. 0,0,0,0,1,1,1,1,1...)

What is Cesaro sum (discrete Heaviside?).

What is Cesaro sum (discrete Heaviside?).

- Cuchulainn
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I hate those kinds of movies.This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/\(\zeta(-1)=-\frac{1}{12}\)

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- katastrofa
**Posts:**7937**Joined:****Location:**Alpha Centauri

The theory of dynamical systems (Lotka-Voltera model, strange attractors, etc.) and practically all modern theories in economy, sociology or biology would not exist if Cauchy's descendants (Poincaré?) dropped the problem at that point - vide limit sets. (I agree that the S=1/2 makes no sense, though.)the [$]n[$]th-term test for divergence:

If [$]\lim_{n\to\infty}a_{n}\ne 0[$] or if the limit does not exist, then [$]\sum _{n=1}^{\infty }a_{n}[$] diverges

Which is what we've got here

Last edited by katastrofa on December 7th, 2018, 2:27 pm, edited 1 time in total.

- FaridMoussaoui
**Posts:**400**Joined:****Location:**Genève, Genf, Ginevra, Geneva

I don't know why but I didn't read the first paragraph about the movie. I started saying what Daniel is speaking about.I hate those kinds of movies.This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/\(\zeta(-1)=-\frac{1}{12}\)

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I understood when I come back to the blog.

As a rule for me, as long as I read a book, I don't watch the movie.

As a rule for me, as long as I wrote the paper, I don't watch the movieI don't know why but I didn't read the first paragraph about the movie. I started saying what Daniel is speaking about.I hate those kinds of movies.

This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/

I understood when I come back to the blog.

As a rule for me, as long as I read a book, I don't watch the movie.

- Cuchulainn
**Posts:**59389**Joined:****Location:**Amsterdam-
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It's just that all these movie are about numbers. I don't understand the fascination. I suppose a movie about [$]\pi[$] is next?I don't know why but I didn't read the first paragraph about the movie. I started saying what Daniel is speaking about.I hate those kinds of movies.

This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/

I understood when I come back to the blog.

As a rule for me, as long as I read a book, I don't watch the movie.

Pi "The story is about a mathematician and the obsession with mathematical regularity contrasts two seemingly irreconcilable entities: the imperfect, irrational humanity and the rigor and regularity of mathematics, specifically number theory"It's just that all these movie are about numbers. I don't understand the fascination. I suppose a movie about [$]\pi[$] is next?I don't know why but I didn't read the first paragraph about the movie. I started saying what Daniel is speaking about.

I hate those kinds of movies.

I understood when I come back to the blog.

As a rule for me, as long as I read a book, I don't watch the movie.

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