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 . But is this correct? Believe it or not, we can rearrange the sum of the natural numbers to be equal to . 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 we say the series converges to .
Consider the following series .This is known as Grandi’s Sum. The partial sums of this series are as follows
if is odd and is k is even. Now the sequence formed by the averages looks like,
We see that . Hence, Grandi’s sum .
Sow let us consider The following sum . We can add 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 hence we conclude .
Now let . then is
1 + 2 + 3 + 4 + 5 + …
– 1 + 2 – 3 + 4 – 5 + …
=0 + 4 + 0 + 8 + 0 + … = 4S.
That is which implies .
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.