1+2+3+4+.... = -1/12

tensaix2j

  

tensaix2j

原来这个series 也曾被印度人研究。。。亦称 Ramanujan_summation

http://en.wikipedia.org/wiki/Ramanujan_summation

The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is also −1/12.[1] In Srinivasa Ramanujan's second letter to G. H. Hardy, dated 27 February 1913, he wrote:

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"[2]http://books.google.com.sg/books?id=Of5G0r6DQiEC&pg=PA53&dq=gratified&redir_esc=y#v=onepage&q=gratified&f=false

mathlim

1 = ♥ + 1 - ♥


1 - 1 + 1 - 1 + 1 - 1 + ... ...

= ( ♥ + 1 - ♥ ) - ( ♥ + 1 - ♥ ) + ( ♥ + 1 - ♥ ) - ( ♥ + 1 - ♥ ) + ( ♥ + 1 - ♥ ) - ( ♥ + 1 - ♥ ) + ... ...

= ♥ + 1 - ♥ - ♥ - 1 + ♥ + ♥ + 1 - ♥ - ♥ - 1 + ♥ + ♥ + 1 - ♥ - ♥ - 1 + ♥ + ... ...

= ♥ + ( 1 - ♥ - ♥ - 1 + ♥ + ♥ ) + ( 1 - ♥ - ♥ - 1 + ♥ + ♥ ) + ( 1 - ♥ - ♥ - 1 + ♥ + ♥ ) + ... ...

= ♥ + 0 + 0 + 0 + ... ...

= ♥

mathlim

Mathlim = Einstein + Mathlim - Einstein
Einstein - Mathlim + Mathlim - Einstein = 0

Mathlim

= Mathlim + 0 + 0 + ... ...

= (Einstein + Mathlim - Einstein) + (Einstein - Mathlim + Mathlim - Einstein) + (Einstein - Mathlim + Mathlim - Einstein) + ... ...

= Einstein + Mathlim - Einstein + Einstein - Mathlim + Mathlim - Einstein + Einstein - Mathlim + Mathlim - Einstein + ... ...

= Einstein + (Mathlim - Einstein + Einstein - Mathlim) + (Mathlim - Einstein + Einstein - Mathlim) + ... ...

= Einstein + 0 + 0 + ... ...

= Einstein

mathlim

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... ...          (1)
S =       1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... ...    (2)
(1) - (2)  
0 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... ...

tensaix2j

Riemann Zeta Function 的故事还没结束呢, 上面只是开胃菜。。。

tensaix2j