(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.

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# Joy of Mathematics

## Conservative field

## An extraordinary sin integral

## On a nested sin sequence

## An interesting limit

## Nested radical inequality

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(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 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 .

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.

Let . Prove that

**Solution**

We note that

Thus,

and the result follows from the sandwich theorem.

Let . Prove that

**Solution**

The LHS is equal to which by AM – GM is less or equal to

where . Since it follows from Bernoulli inequality that .