Amazing..!! How can this even be possible ??

Prove that:

1+2+3+4+5+6+.... up to infinity = -1/12 (negative of one-twelfth)

The above identity is used in many derivations of physics. I have a mathematical proof of the following, but please provide a logical one..

#Identity #Infinity #FunFacts #Mathsfun #Mathsclub

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

It took me 1 day to personally find out the mistakes that the university professors have done. Those people forgot the basic conditions required for treating a series as converging infinite series. In short this was just one stupid act of getting fame

Achint Gupta - 7 years, 3 months ago

are u sure???....could u show the proof???

Tanya Gupta - 7 years, 3 months ago

Please see my reply to Achint Gupta..

Rushi Rokad - 7 years, 3 months ago

Saw it...The numberphile video was amazing!!...Thanx!!

Tanya Gupta - 7 years, 2 months ago

hows this possible???? I mean sum of positive natural numbers is negative !!!!!! How ???

Achint Gupta - 7 years, 3 months ago

Let S1 = 1 + -1 + -1 + 1 + -1 .....up to infinity = 1/2 (this can be proved by using the sum of infinite GP and even by simpler method. Let S2 = 1 -2 +3 -4 + 5 -6...up to infinity S2 + S2 = (1 -2 +3 -4....)+ (0 +1 -2 +3..) =S1 Therefore, 2 S2 = S1 = 1/2 Therefore, S2=1/4

Let S3 = 1 + 2 + 3 + 4 + 5.... Now, S3 - S2 = (1 + 2 + 3 +4...) -(1 -2 +3 -4..) = 4 + 8 + 12 +16..... = 4 (1+ 2 +3 +4...) = 4 (S3) Therefore, 3 (S3) = - S2 S3 = -1/12...

Sorry,I could not present it properly here.... For a better explanation , watch Numberphile on Youtube - https://www.youtube.com/watch?v=w-I6XTVZXww&list=TLB3KTqn9SZmDUThXfkDzwmDz8PRN0U-K-

Rushi Rokad - 7 years, 3 months ago

This stuff can never ever be proved by "using sum of infinite GP or by any other simple maths ". But I thank u for that video link which shows us but i cant understand it as it involves some higher mathematics which i haven't yet studied

Achint Gupta - 7 years, 3 months ago

@Achint Gupta – I got really something interesting for u After reading through the below link, u will find that such things aren't possible .

http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/

Achint Gupta - 7 years, 3 months ago

@Achint Gupta – Dat was Nice...

Rushi Rokad - 7 years, 3 months ago

@Achint Gupta – why can't we use sum of infinite GP..and take common ratio as -1 . we get the sum of the series as 1/2 .. ???

Brij Rokad - 7 years, 3 months ago

@Brij Rokad – To use sum of infinite GP we have a condition that | r | < 1 , where r denotes the common ratio and " | | " is the modulus function

Achint Gupta - 7 years, 3 months ago

@Achint Gupta – You can do it as such...

S1 = 1 + -1 +1 + -1 ...... S1 + S1 =( 1 + -1 + 1 + -1 + ...)+ ( 1 + -1 + 1 + -1...) 2 (S1) = 1 S1 = 1/2 ..Hence...

Rushi Rokad - 7 years, 3 months ago

hey, that was grt proof.

Anand Narayan - 7 years, 2 months ago

@Anand Narayan – Not mine..

Rushi Rokad - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$