Infinity is tricky to deal with

Calculus Level 3

I will attempt to prove that ∞ = 1 or ∞ = − 2 \infty = 1 \text{ or }\infty= -2 . In which of these steps did I first make a flaw in my logic?

Step 1: Using the algebraic identities , we have

1 + 2 + 3 + ⋯ + n = 1 2 n ( n + 1 ) 1 3 + 2 3 + 3 3 + ⋯ + n 3 = 1 4 n 2 ( n + 1 ) 2 \begin{array} { l l l l l l } 1 &+ 2 &+ 3 &+ &\cdots &+ &n = \dfrac12 n(n+1) \\ 1^3 &+ 2^3 &+ 3^3 &+ &\cdots &+ &n^3 = \dfrac14 n^2(n+1)^2 \\ \end{array}

Step 2: Connecting these two equations to get

( 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: Since 1 + 2 + 3 + ⋯ = ∞ 1 + 2 + 3 + \cdots = \infty and 1 3 + 2 3 + 3 3 + ⋯ = ∞ 1^3 + 2^3 + 3^3 + \cdots = \infty , then

1 + 2 + 3 + ⋯ + ∞ = 1 3 + 2 3 + 3 3 + ⋯ + ∞ 3 . 1+ 2 +3 + \cdots + \infty = 1^3 + 2^3 + 3^3 + \cdots + \infty^3 .

Step 4: We can rewrite the equation in Step 3 as

1 2 ∞ ( ∞ + 1 ) = ( 1 2 ∞ ( ∞ + 1 ) ) 2 . \dfrac12 \infty(\infty+1) = \left( \dfrac12 \infty(\infty+1) \right)^2 .

Step 5: Canceling the common factors gives

1 = 1 2 ∞ ( ∞ + 1 ) . 1 = \dfrac12 \infty (\infty + 1) .

Step 6: Solving this quadratic equation gives ∞ = 1 \infty =1 or ∞ = − 2 \infty = -2 .

Step 5 Step 2 Step 4 Step 3 Step 6

Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Let us look at the steps carefully.

  • Step I uses two summation identitites about natural numbers. Looking at the wiki, this seems fine.
  • Step II can be justified using regular algebraic methods.
  • Step III seems to be something fishy. Actually, it is ambiguous in several levels.

The first line says

Since 1 + 2 + 3 + ⋯ = ∞ 1 + 2 + 3 + \cdots = \infty and 1 3 + 2 3 + 3 3 + ⋯ = ∞ 1^3 + 2^3 + 3^3 + \cdots = \infty , then

and then the second line says

1 + 2 + 3 + ⋯ + ∞ = 1 3 + 2 3 + 3 3 + ⋯ + ∞ 3 . 1+ 2 +3 + \cdots + \infty = 1^3 + 2^3 + 3^3 + \cdots + \infty^3 .

For some reason, we have taken for granted that 1 + 2 + 3 + ⋯ 1 + 2 + 3 + \cdots is the same as 1 + 2 + 3 + ⋯ + ∞ 1+ 2 +3 + \cdots + \infty

This is not really true. The sum 1 + 2 + 3 + ⋯ 1 + 2 + 3 + \cdots runs over all the natural numbers. And strangely 1 + 2 + 3 + ⋯ + ∞ 1+ 2 +3 + \cdots + \infty also includes ∞ \infty . But ∞ \infty is not really a natural number. That is where this whole argument breaks down. Recall that in Step I, we claimed that the identities are only about natural numbers .

Also, it is not clear in what sense 1 + 2 + 3 + ⋯ = 1 3 + 2 3 + 3 3 + ⋯ 1 + 2 + 3 + \cdots = 1^3 + 2^3 + 3^3 + \cdots is true. What does it mean for two infinite series to be equal? Should we compare term by term? Should we compare partial sums? Should we try to subtract in some sense and get 0?

The symbol ∞ \infty is usually dependent on context.

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Prince Loomba
Jan 20, 2017

Step 3 compares 2 infinities, and says they are equal, but they are not. 2 infinities are not necessarily equal

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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