On a polygamma product series

Let denote the trigamma function. Prove that

(Seraphim Tsipelis)

Solution

We are quoting several lemmata in order to prove this result.

Lemma 1: It holds that .

Proof:

Lemma 2: It holds that .

Proof:

Lemma 3: We define with .

Lemma 4: It holds that .

Proof:

and the result follows.

Lemma 5: It holds that .

Lemma 6: It holds that .

Proof:

The proof relies on Multiple Zeta functions. Since it holds that

then we have that

Now returning to the question in hand we have that

and what remains in order to complete the exercise is the evaluation of those two last Euler sums. We proceed with the second one.

while for the first one

Combining the results we get what we want.

A series with least common multiple

Let be a strictly increasing sequence of positive numbers. For all denote as the least common multiple of the first terms of the sequence. Prove that , as , the following sum converges

Solution

This is a result due to Paul Erdös stating that if are natural numbers such that then

and the original question follows since the sum we seek is less or equal to .

However, we are presenting another proof. Denote as the average order of the numbers , i.e.,

For any we have where is the product of primes not present in the factorization of . Note that are squarefree integers. Note also that it may be an empty product, i.e., . Then

It is easy to see (and show by induction) that so we have

Hence, Consequently, we have

So the sum of reciprocals of converges. Then, by Cesàro summation, we see that

also converges.

Divisibility

Prove that the product of consecutive positive integers divides .

Solution

The number of different combinations of over is of course an integer and equals to

and the result follows.

Any questions?

Inequality in acute triangle

Let be an acute triangle. Prove the following inequality

Solution

The solution can be found at cut -the – knot

Inequality in a triangle

Given a triangle let denote the median points of the sides respectively. Prove that