Firstly, let's look at the infinite sum (it's called Grandi's Series)
What is its value? Well, it turns out to be ...
So why is that true?
Well, let's call .Then
! So and
Now let's look at the sum
(The is the Dirichlet eta function )
Interestingly, the value of this sum is actually !
Proof 1:We can rewrite the sum as
Proof 2:Let's look at the infinite geometric series:
for x less than 1.
Now differentiating both sides we get:
Now substituting we get that
Now let's set our task to find the value of
Well, the sum is obviously divergent, but it's value can also be calculated as !!! Isn't it amazing? Well, maybe you think that I am some kind of lunatic, but nevertheless, I will show you the proof.
First, let me briefly introduct you to the Riemann zeta function.It is defined like this:
So what we are seeking for is .
The following observations were made by Leonhard Euler ( he didn't use the same notation):
That reminds us of the good ol'
And indeed if we set we would get that
Note:The sum is divergent, but Euler summable! The method we used to calculate is called analytic continuation which extends the domain of a given function. We set, where the sum is divergent, but we used a formula for the values in the domain.Also, the whole post was inspired by Numberphile.