Question

Sum upto infinity of divergent series

Usually i had thought when we have divergent series the sum upto infinity is infinity..

but
1+2+3+4... = -1/12 :P..
now i looked this up, and of course there are proofs available, but logic disproves it, like for eg i can use simple mathematics and say
1+2+3.. > 1+2.. so all those things say the result is wrong.. also i can do

1+2+3+4 ... = -1/12
1+ (1+1)+(1+2)+(1+3)= -1/12
(1+1+1+1....) +(1+2+3+ ... ) = -1/12
(1+1+1+1..)+ (-1/12)= -1/12
hence (1+1+1+1.....) = 0

so i don't want u guys to prove it :p..

Answer 1

can someone help me understand why our DISPROOFS using simple maths is invalid?

@Vincent-Lyon.Fr @AllTehMaffs @phi @cwrw238 @ajprincess @Compassionate

Answer 2

your assumption 1+2+3+4+....=-1/12 is invalid

Answer 3

its not my assumption @atlas its proved to be true.. believe me.. i don't believe it.. but i did research and everywhere its true..

Answer 4

here is one source numberphile :D

http://kottke.org/14/01/the-sum-of-all-positive-integers

Answer 5

it is invalid

Answer 6

:-/ could you elaborate? or go through the site.. before just bluntly saying its invalid? :P.. yes i know it looks invalid..

Answer 7

let me check put

Answer 8

thank you :D

Answer 9

the site does not open up here....can you give me any other link

Answer 10

ok here is a wiki link

http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

Answer 11

watch:
http://www.youtube.com/watch?v=w-I6XTVZXww&feature=c4-overview&list=UUoxcjq-8xIDTYp3uz647V5A

Answer 12

This formula actually exists in a book, like the guy in the video said. However, we all know this sum is incorrect because it appears in a physics book XD

Answer 13

The person in the video is adding two divergent series terms by terms. You cannot do that. In fact you can arrange any convergent alternating series to converge to any number.

Answer 14

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

Answer 15

Say how they prove in the above link that 1=0

Answer 16

ok i saw the proof and it seemed mathematically correct but the proof was obtained by rearranging the terms of a divergent series which did not feel right....:

Answer 17

It is like saying:
S = 1+2+3+4+....
S= 1+2+3+......
Subtracting both
0 = 1+1+1+1......
and so 0 = infinity

Answer 18

Indeed, this answer from Quora agrees with @eliassaab http://www.quora.com/Mathematics/Theoretically-speaking-how-can-the-sum-of-all-positive-integers-be-1-12
They explain that zeta(-1)/12 but that's a result of something I don't seem to understand.

Answer 19

ok guys thanks for taking interest.. i have realized that the word "summation" itself is at stake here,
so this summation is not like adding 1 apple + 2 apple + 3 apple +.... upto infinity .. cause if we did.. we know we can't get -1/12 apples :D..

So i guess this is something that will be applied in string theory and quantum mechanics only.. ll do further research to see it.. thanks again guys :D

Answer 20

@eliassaab
that i think was just a simple proof.. there are more elaborate proofs.. but my original question was not to understand the proof, but to understand the meaning :P

Answer 21

@Mashy, in logic if statement A is false then for any statement B, the statement
\[
A \implies B
\]
is true.
If you use a false method, you can prove anything you want.

Answer 22

no no.. they just used that method in laymans terms..

it has been proved using some complicated math.. look up Ramanujan series.. the great indian scientist has ALSO proved it.. its called as Ramanujan summation..
so don't bother about the VALIDITY.. or its proof.. its right.. its been accepted officially.. i just wanted to understand it!

Answer 23

I am hoping that you can understand it. Good luck.

Answer 24

thank you :)

Answer 25

@Mashy but I understood your question and it was a good question to ask.
However, I am not so deep into mathematics but I hope you get the right person to answer the question. Besides I think this link might be useful http://math.arizona.edu/~cais/Papers/Expos/div.pdf and you may like to follow it up :)