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 3 + 2 3 + 3 3 + ⋯ + n 3 = = 2 1 n ( n + 1 ) ( 2 1 n ( n + 1 ) ) 2
Step 2 : There is a relationship between the numbers 1 + 2 + 3 + ⋯ + n and 1 3 + 2 3 + 3 3 + ⋯ + n 3 ,
( 1 + 2 + 3 + ⋯ + n ) 2 = 1 3 + 2 3 + 3 3 + ⋯ + n 3
Step 3 : Take the limit to both sides of the equation,
n → ∞ lim ( 1 + 2 + 3 + ⋯ + n ) 2 = n → ∞ lim ( 1 3 + 2 3 + 3 3 + ⋯ + n 3 )
Which is equivalent to
( 1 + 2 + 3 + ⋯ ) 2 = 1 3 + 2 3 + 3 3 + ⋯
Step 4 : Since 1 3 + 2 3 + 3 3 + ⋯ → ∞ and 1 + 2 + 3 + ⋯ → ∞ , then 1 3 + 2 3 + 3 3 + ⋯ = 1 + 2 + 3 + ⋯ ,
( 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
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Not exactly. You can't write the equation of infinity = infinity. It's just mathematically wrong.
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More to the point, writing down a limit lim n → ∞ implies that said limit exists (and is finite). Neither of these two limits exist. It is true to say that both sides diverge to ∞ , but that statement would be written differently.
We could argue on this point, Comrade Pi. As usual, what is "mathematically wrong" depends on your definitions. It is perfectly fine to write ∞ = ∞ 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.
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Well, by convention, we don't "work on" Extended real number line, don't we?
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@Pi Han Goh – Well, if you attend a math talk and the speaker writes ∞ = ∞ on the board, don't tell him that this is "just mathematically wrong" ;)
Yes that's right.
Your first error is in the title - it definitely "should" not be equal to − 1 2 1 , 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!!
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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 + . . . . ? 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.