Home » Uncategorized

# Category Archives: Uncategorized

## Homogeneity of inequality

Let . Prove that:

**Solution**

Due to homogeneity we may assume . Thus there exist positive such that

Hence,

## Sinc series

The sinc function is defined as:

Prove that for any couple of real numbers in the following result holds:

**Solution**

Since

due to the addition formulas for the sine and cosine functions it is enough to prove the equality

which is an immediate consequence of the Fourier series by integrating the sawtooth wave function;

Hence,

## Continuous binomial integral

Let . Prove that

**Solution**

First of all,

due to the well known formulae and .

However using the residue theorem we get that

Thus,

due to the binomial theorem and the well known fact

## Trigonometric inequality on a triangle

Let be a triangle. Prove that

**Solution**

The function is convex, thus:

The result follows. In fact, something a bit stronger holds. Let denote the inradius , the circumradius and the semiperimeter. Then,

Indeed,

in view of the known identities

(1)

(2)

(3)

## A trigonometric limit

Evaluate the limit:

**Solution**

Let us begin with the Fourier series of which is of the form:

Hence

(1)

Integrating ( 1 ) we get that:

The Riemann – Lebesgue Lemma implies that

and thus .