But it should be equal to -1/12

Calculus Level 4

Here's my proof that the sum of all the natural numbers is equal to 1.
In which of thse steps did I first make a flaw in my logic?

Step 1 : Recall the two algebraic identities ,

1 + 2 + 3 + + n = 1 2 n ( n + 1 ) 1 3 + 2 3 + 3 3 + + n 3 = ( 1 2 n ( n + 1 ) ) 2 \begin{aligned} 1 + 2 + 3 + \cdots + n &=& \dfrac12n(n+1) \\ 1^3 + 2^3 + 3^3 + \cdots + n^3 &=& \left( \dfrac12n(n+1) \right)^2 \\ \end{aligned}

Step 2 : There is a relationship between the numbers 1 + 2 + 3 + + n 1 + 2 + 3 + \cdots + n and 1 3 + 2 3 + 3 3 + + n 3 1^3 + 2^3 + 3^3 + \cdots + n^3 ,

( 1 + 2 + 3 + + n ) 2 = 1 3 + 2 3 + 3 3 + + n 3 (1 + 2 + 3 + \cdots + n )^2 = 1^3 + 2^3 + 3^3 + \cdots + n^3

Step 3 : Take the limit to both sides of the equation,

lim n ( 1 + 2 + 3 + + n ) 2 = lim n ( 1 3 + 2 3 + 3 3 + + n 3 ) \lim_{n\to\infty} (1 + 2 + 3 + \cdots + n )^2 = \lim_{n\to\infty} (1^3 + 2^3 + 3^3 + \cdots + n^3)

Which is equivalent to

( 1 + 2 + 3 + ) 2 = 1 3 + 2 3 + 3 3 + (1+2+3 + \cdots )^2 = 1^3 +2^3 + 3^3 + \cdots

Step 4 : Since 1 3 + 2 3 + 3 3 + 1^3 +2^3 + 3^3 + \cdots \rightarrow \infty and 1 + 2 + 3 + 1 +2 + 3 + \cdots \rightarrow \infty , then 1 3 + 2 3 + 3 3 + = 1 + 2 + 3 + 1^3 +2^3 + 3^3 + \cdots = 1 +2 + 3 + \cdots ,

( 1 + 2 + 3 + ) 2 = 1 + 2 + 3 + ( 1 + 2 + 3 + ) ( 1 + 2 + 3 + ) = 1 + 2 + 3 + ( 1 + 2 + 3 + ) ( 1 + 2 + 3 + ) = ( 1 + 2 + 3 + ) 1 + 2 + 3 + = 1 \begin{aligned} (1+2+3 + \cdots )^2 &=& 1 +2 + 3 + \cdots \\ (1+2+3 + \cdots )(1+2+3 + \cdots ) &=& 1 +2 + 3 + \cdots \\ \cancel{(1+2+3 + \cdots )}(1+2+3 + \cdots ) &=& \cancel{(1+2+3 + \cdots )} \\ 1+2+3 + \cdots &=& 1 \end{aligned}

Step 1 Step 2 Step 3 Step 4 No flaw at all, your logic is perfect!

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2 solutions

Sherwin D'souza
Mar 18, 2016

In step 3, after taking the limit, how can one be sure that ( 1 + 2 + 3 + 4 + . . . . ) 2 = 1 3 + 2 3 + 3 3 + 4 3 + . . . . ? (1+2+3+4+....)^2 = 1^3+2^3+3^3+4^3+....? Since the last terms could be any distinct numbers too, But there is no last term here, and so the equality itself is indeterminate.

I hope I'm correct, if there are any flaws notify me.

Not exactly. You can't write the equation of infinity = infinity. It's just mathematically wrong.

Pi Han Goh - 5 years, 2 months ago

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More to the point, writing down a limit lim n \lim_{n\to\infty} implies that said limit exists (and is finite). Neither of these two limits exist. It is true to say that both sides diverge to \infty , but that statement would be written differently.

Mark Hennings - 5 years, 2 months ago

We could argue on this point, Comrade Pi. As usual, what is "mathematically wrong" depends on your definitions. It is perfectly fine to write = \infty = \infty when you work on the "Extended real Number Line" as long as you are careful with your arithmetic operations. The real error occurs in Step 4.

Otto Bretscher - 5 years, 2 months ago

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Well, by convention, we don't "work on" Extended real number line, don't we?

Pi Han Goh - 5 years, 2 months ago

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@Pi Han Goh Well, if you attend a math talk and the speaker writes = \infty = \infty on the board, don't tell him that this is "just mathematically wrong" ;)

Otto Bretscher - 5 years, 2 months ago

Yes that's right.

Sherwin D'souza - 5 years, 2 months ago
Mark Hennings
Mar 19, 2016

Your first error is in the title - it definitely "should" not be equal to 1 12 -\tfrac{1}{12} , unless you are going to get into elementary particle physics and talk about "zeta function regularized sums".

Ohhhhh!!!!! I didn't ask you to look at the title! ahahahahah

Upvoted anyway!!

Pi Han Goh - 5 years, 2 months ago

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