Home » Posts tagged 'Series'

# Tag Archives: Series

## 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,

## Double summation

Let be a real number. Evaluate the series:

**Solution**

Since the summands are all positive , we can sum by triangles. Thus,

where is the Riemann zeta function.

## Floor series

Let denote the floor function. Evaluate the series

**Solution**

First of all we note that and are never squares. Thus, there exists a positive integer such that

It is easy to see that and thus we conclude that

Now is equal to the even number if-f

Hence, since the series is absolutely convergent we can rearrange the terms and by noting that the finite sums are telescopic , we get that:

## Double “identical” series

Compute the series

**Solution**

Successively we have:

## On Euler’s totient function series

Let denote Euler’s totient function. Prove that for it holds that:

where stands for the Riemann zeta function.

**Solution**

Well by Euler’s product we have,

thus,

(1)

and

(2)

Combining we get the result.

**Note: **It also holds that