## Harmonic zeta tail series

Let denote the – th harmonic number. Evaluate the series

Solution

Successively , we have:

## Inverse zeta(3) limit

Evaluate the limit

We are proving that the limit is . Indeed , one has:

If then

whereas if then the above limit is . Thus:

## Conservative field

(i) Let such that . Let be a normal area. Prove that

(ii) Examine if is a conservative field. If so, fiend a force of it.

(i) Using Green’s theorem we have that

(ii) Yes, it is. We note that

and of course a force is .

## An extraordinary sin integral

I was surfing the net today and I fell on this cute integral

I have seen integrals of such kind before like for instance this .In fact something  more general holds

where .

Solution

The original integral does not fall into this category which is a real shame. Yet it does have a closed form and it does not contain an in its final answer. Strange, huh? So similar but so different at the same time these two integrals. A sign changes everything.

We begin by exploring the integral

Manipulating the integral ( substitutions and known Gaussian results) reveals that

where . Taking the imaginary part of the last expression we get that

and this is the final answer. See, no !. Of course we can also extract the real part and calculate the corresponding integral involving .

## On a nested sin sequence

Consider the sequence defined recursively as

Prove that .

Solution

Lemma: If is a sequence for which then

Proof: In Stolz theorem we set and .

It is easy to see that is is monotonically decreasing to zero. Moreover, an application of L’Hospital’s rule gives

Therefore

Now, due to the lemma we have and the result follows.

Remark : The asymptotic now follows to be .

Problem: Find what inequality should satisfy such that the series

converges.