Summing the Natural Numbers Produces Unexpected Results

Summing the Natural Numbers Produces Unexpected Results

Sum

Claim:

The sum of the natural numbers S = 1 + 2 + 3 + 4 + … = -1/12.

Informal Proof:

Let’s define a sum of natural counting numbers as follows:

S = 1 + 2 + 3 + 4 + 5 + …

At first glance the answer is either: “there is no sum (no finite number)” or “the sum is infinity”.

Fortunately Leonard Euler had a bit more insight to this sum and the answer is quite revealing.

There are many ways of drawing the same conclusion that Euler did as to the solution to this sum, but what follows is the easiest I’ve been able to come up with to explain it.

First let’s look at a related series called the Taylor Series:

T = 1 – 2 + 3 – 4 + 5 …

Note: In any series one is allowed to add (or subtract) arbitrary 0’s without affecting the sum. For example
T = 1 – 2 + 3 – 4 + 5 … = 0 + 1 – 2 + 3 – 4 + 5… = 0 + 0 + 1 – 2 + 3 – 4 + 5…

Let’s sum 4 Taylor Series to see if this helps figure out what the sum of the Taylor Series is.

   T = 1 - 2 + 3 - 4 + 5 - 6 + 7 ...
   T = 0 + 1 - 2 + 3 - 4 + 5 - 6 ...
   T = 0 + 1 - 2 + 3 - 4 + 5 - 6 ...
   T = 0 + 0 + 1 - 2 + 3 - 4 + 5 ...
 +
 ---------------------------------------
  4T = 1 + 0 + 0 - 0 + 0 - 0 + 0 ...
  4T = 1
[A] T = 1/4

Wait!: “That’s totally unfair” I can imagine many people saying in protest of my magically ‘knowing’ to sum 4 T’s and arrange the terms as I did. So how did I happen to know to sum 4 T’s and arrange the terms that way?! Mathematics has parts which are ‘trial and error’. Mathematicians commonly manipulate sequences while maintaining equality and ‘play’ with them. They ‘play’ to see if they can tease deeper meanings from the sequences they’re studying. I don’t claim to have spent the time coming up with this particular rearrangement, I found it on the internet. I like to call it “using my ‘Google brain'”.

So armed with the fact that T = 1/4, let’s reexamine the sum of natural numbers:

[B] S = 1 + 2 + 3 + 4 + 5 + ...

Multiplying both sides by 4 we get

4S = 4*(1 + 2 + 3 + 4 + 5 + ...) = 4 + 8 + 12 + 16 + 20 + ...

Recall that we’re free to pad a series with 0’s without changing it’s value.

4 + 8 + 12 + 16 + 20... = 0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + 0 + 20 + ...

So we have the result

[C] 4S = 0 + 4 + 0 + 8 + ...

Subtracting results [B] – [C] we get

  S = 1 + 2 + 3 + 4 + 5 + ...
 4S = 0 + 4 + 0 + 8 + ...
 -
 -----------------------------------------
-3S = 1 - 2 + 3 - 4 + 5 - ...

Wait! Again: how did I know to take S – 4S to get the Taylor Series and pad the 4S term such as I did? Again, it was trial and error and trial and error I didn’t perform but a result I found by using my “Google Brain”.

Examining the right hand side of the equation, we recognize it to be the Taylor Series, we’ve examined previously. From result [A], we know that the we can the summation of the Taylor Series is 1/4. This allows us to replace the right hand side with 1/4 yielding:

-3S = 1/4

multiplying both sides by -1/3 we get

S = - 1/12

Conclusion:

The summation of the natural numbers (1 + 2 + 3 + 4 + …) is – 1/12th.

Discussion:

The result that the sum of the natural numbers is shocking and completely unintuitive. Here is a chart of what our intuition says about this sum and what the actual result is:

Intuition Actual Result
The sum is a huge positive number (infinity). The result is a small negative number.
The sum is a natural number. The result is a rational number (fraction).

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