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# Tag Archives: Series

## Floor series

Let denote the floor function. Evaluate the series

Solution

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:

## Double “identical” series

Compute the series

Solution

Successively we have:

## On Euler’s totient function series

Let denote Euler’s totient function. Prove that for it holds that:

where stands for the Riemann zeta function.

Solution

Well by Euler’s product we have,

thus,

(1)

and

(2)

Combining we get the result.

Note: It also holds that

## A hypergeometric series

Let such that . Evaluate the series:

Solution

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:

## Series of zeta sum

Let denote the Riemann zeta function. Evaluate the series:

Solution

Let . We are proving the more general result.

where denotes the digamma function.

First of all, we note that:

Thus,

due to the reflection formula .

Side note: If then the sum equals .

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