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bearish
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Re: Universe Tea - Blog 2018

December 6th, 2018, 9:49 pm

I have seen this movie before (literally). Go down this path and discover that the sum of all natural numbers equals -1/12.
 
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Collector
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Re: Universe Tea - Blog 2018

December 6th, 2018, 11:02 pm

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

बुत् बुद्द सयस 
 
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Cuchulainn
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Re: Universe Tea - Blog 2018

December 7th, 2018, 12:15 am

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?)
 
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ppauper
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Re: Universe Tea - Blog 2018

December 7th, 2018, 12:40 am

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
 
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katastrofa
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Re: Universe Tea - Blog 2018

December 7th, 2018, 8:10 am

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 :-)
 
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ppauper
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Re: Universe Tea - Blog 2018

December 7th, 2018, 8:39 am

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
 
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Cuchulainn
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Re: Universe Tea - Blog 2018

December 7th, 2018, 10:40 am



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.

Image
 
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FaridMoussaoui
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Re: Universe Tea - Blog 2018

December 7th, 2018, 11:35 am

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

बुत् बुद्द सयस 
This is an interesting post by Wolfram on that Ramanujan formula: https://blog.stephenwolfram.com/2016/04 ... ramanujan/
 
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Cuchulainn
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Re: Universe Tea - Blog 2018

December 7th, 2018, 12:51 pm

(.. 0,0,0,0,1,1,1,1,1...)
What is Cesaro sum (discrete Heaviside?).
 
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Cuchulainn
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Re: Universe Tea - Blog 2018

December 7th, 2018, 1:10 pm

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

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. 
 
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katastrofa
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Re: Universe Tea - Blog 2018

December 7th, 2018, 1:33 pm

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.
 
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FaridMoussaoui
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Re: Universe Tea - Blog 2018

December 7th, 2018, 2:18 pm

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

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.
 
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Collector
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Re: Universe Tea - Blog 2018

December 7th, 2018, 2:37 pm


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
 
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Cuchulainn
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Re: Universe Tea - Blog 2018

December 7th, 2018, 2:39 pm


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?
 
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Collector
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Re: Universe Tea - Blog 2018

December 7th, 2018, 2:40 pm



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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