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

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?)

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

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

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

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.

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

बुत् बुद्द सयस 

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

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

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

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. 

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.)

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

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.

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?