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# Tag Archives: Series

## 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 .

## Harmonic sum with reciprocal central binomial coefficient

Let denote the – th harmonic number. Prove that

where denotes the Catalan’s constant.

**Solution**

We begin with a lemma:

* Lemma: *Let denote the dilogarithm function. It holds that

*Proof: *It is well known that

(1)

Setting we have that:

Setting we have that:

However,

where is the inverse tangent function. It now follows that

in view of the well known series .

Substracting the above relations we get the proof of the lemma.

* Theorem: *Let . It holds that

Setting we have that . The result now follows immediately using the lemma above as well as the fact that .

## On Vacca’s formula

Let denote the Euler Mascheroni constant. Prove that

where denotes the floor function.

**Solution**

We have successfully

## A beautiful Gamma series

Let denote the Gamma function. Prove that

**Solution**

The Taylor series is

Hence,

and the result follows.

## Series with number of bits ‘1’

Let count the number of bits ‘1’ in the binary representation of . Calculate the series

**Solution**

For , consider the function:

We have:

because the summands in the double sum are all non-negative numbers and allow us to perform the double sum in any order we want.

Notice iff for some integer . Thus: