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bearish
Posts: 4571
Joined: February 3rd, 2011, 2:19 pm

### Re: Universe Tea - Blog 2018

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

Collector
Posts: 4172
Joined: August 21st, 2001, 12:37 pm

### Re: Universe Tea - Blog 2018

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

बुत् बुद्द सयस

Cuchulainn
Posts: 58957
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Universe Tea - Blog 2018

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

Find S
1/2
indeed, IF S EXISTS THEN
S-1 = -1 +1 -1 +1 .... (forever) =-S
so 2S=1
It's Grandi's series and its Cesaro sum is 1/2.
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?)

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Universe Tea - Blog 2018

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

katastrofa
Posts: 7569
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Universe Tea - Blog 2018

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

Find S
1/2
indeed, IF S EXISTS THEN
S-1 = -1 +1 -1 +1 .... (forever) =-S
so 2S=1
Indeed if S-1 = -S, then S = 1/2. Cauchy and Weierstrass turn in their graves

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Universe Tea - Blog 2018

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

Cuchulainn
Posts: 58957
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Universe Tea - Blog 2018

1/2
indeed, IF S EXISTS THEN
S-1 = -1 +1 -1 +1 .... (forever) =-S
so 2S=1
Indeed if S-1 = -S, then S = 1/2. Cauchy and Weierstrass turn in their graves
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,...).
Maybe they did not think about things they did not think about.

FaridMoussaoui
Posts: 369
Joined: June 20th, 2008, 10:05 am

### Re: Universe Tea - Blog 2018

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

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

Cuchulainn
Posts: 58957
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Universe Tea - Blog 2018

(.. 0,0,0,0,1,1,1,1,1...)
What is Cesaro sum (discrete Heaviside?).

Cuchulainn
Posts: 58957
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Universe Tea - Blog 2018

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

बुत् बुद्द सयस
This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/
I hate those kinds of movies.

katastrofa
Posts: 7569
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Universe Tea - Blog 2018

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

FaridMoussaoui
Posts: 369
Joined: June 20th, 2008, 10:05 am

### Re: Universe Tea - Blog 2018

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

बुत् बुद्द सयस
This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/
I hate those kinds of movies.
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 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.

Collector
Posts: 4172
Joined: August 21st, 2001, 12:37 pm

### Re: Universe Tea - Blog 2018

This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/
I hate those kinds of movies.
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 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 movie

Cuchulainn
Posts: 58957
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Universe Tea - Blog 2018

This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/
I hate those kinds of movies.
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 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.
It's just that all these movie are about numbers. I don't understand the fascination. I suppose a movie about $\pi$ is next?

Collector
Posts: 4172
Joined: August 21st, 2001, 12:37 pm

### Re: Universe Tea - Blog 2018

I hate those kinds of movies.
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 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.
It's just that all these movie are about numbers. I don't understand the fascination. I suppose a movie about $\pi$ is next?
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"
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