Summing the Natural Numbers

Infinite sums are weird. Some converge and some do not. One might think it easy to conclude that if we sum the natural numbers, we would get 1 + 2 + 3 + ... = \infty. But is this correct? Believe it or not, we can rearrange the sum of the natural numbers to be equal to \frac{-1}{12}. We will give the proof of this, but first a few preliminaries.

We will follow the development of Numberphile’s video on the topic.

A series is said to be Cesáro summable if the sequence formed by the average of its partial sums converges. If this sequence converges to A we say the series converges to A.

Consider the following series 1 -1 + 1 - 1 + ....This is known as Grandi’s Sum. The partial sums of this series are as follows

S_k = 1 if k is odd and S_k = 0 is k is even. Now the sequence formed by the averages looks like,

t_{n} = 1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, ...

We see that  \lim_{n\to\infty}t_n = 1/2. Hence, Grandi’s sum S_1 = 1/2.

Sow let us consider The following sum S_2 = 1 -2 +3 -4 + .... We can add S_2 to itself in a clever way.

1 – 2 + 3 – 4 + 5-…

+    1 – 2 + 3 – 4 + …

=1 – 1 + 1 – 1 + 1 – … = 1/2

So this really means that 2 S_2 = 1/2 hence we conclude S_2 = 1/4.

Now let S = 1 + 2 + 3 + 4 + .... then S-S_2 is

1 + 2 + 3 + 4 + 5 + …

– 1 + 2 – 3 + 4 – 5 + …

=0 + 4 + 0 + 8 +  0 + … = 4S.

That is S-1/4 = 4S which implies S = -1/12.

There is another development of this fact using the analytical continuation of the Riemann-Zeta function but we won’t go into that here. This result also appears in physics when describing the Casmir Effect. For more on this check out “Being pushed around by empty space: The Casmir Effect” on gravityandlevity.wordpress.com.

But what is really going on? This is actually due to the Riemann Rearrangement theorem. Given a divergent series and any real number there is a rearrangement of that series that will converge to that number. It just turns out that the one we explained is really important and arises in a few places in the sciences.

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3 thoughts on “Summing the Natural Numbers

  1. That is true that divergent series can be rearranged to sum to any real number. This one just happens to be special if you check out the analytical continuation of the Riemann-Zeta function and/or the Casmir effect it is verified for now. As I said in the post we used the Cesar sum to obtain the result.

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    1. Sorry, I confused the two series 1-1+1… and 1+2+3…

      One request, replace, “Believe it or not, the sum of the natural numbers is actually equal to -1/12″ with ” Believe it or not, a kind of sum of the natural numbers is actually equal to -1/12″

      My earlier comment is non-sense, so I would request you to delete it.

      Like

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