Home » Posts tagged 'Series'
Tag Archives: Series
Let denote the floor function. Evaluate the series
First of all we note that and are never squares. Thus, there exists a positive integer such that
It is easy to see that and thus we conclude that
Now is equal to the even number if-f
Hence, since the series is absolutely convergent we can rearrange the terms and by noting that the finite sums are telescopic , we get that:
Compute the series
Successively we have:
Let denote Euler’s totient function. Prove that for it holds that:
where stands for the Riemann zeta function.
Well by Euler’s product we have,
Combining we get the result.
Note: It also holds that
Let such that . Evaluate the series:
Lemma 1: For the functions , it holds that:
Lemma 2: It holds that:
Proof: Simple calculations using Lemma reveal the identity.
Lemma 3: Using Lemma 2 it holds that
and as a consequence
Then, successively we have that:
Let denote the Riemann zeta function. Evaluate the series:
Let . We are proving the more general result.
where denotes the digamma function.
First of all, we note that:
due to the reflection formula .
Side note: If then the sum equals .