## On a sequence and series

Let be a sequence of real numbers , and .

1. Prove that if converges and exists ( finite or infinite ) then .
2. Give an example where , converges but does not exist.
3. Give an example of a decreasing sequence , but diverges.
4. Prove that if converges and is decreasing then

Solution

1. Let be the sequence of partial sums , then . It follows from Cesaro that . Hence

From the assumption we have that

Substracting these two we have that

But since it follows from Cesaro and the uniqueness of the limit that the last sum tends to .

2. One such example could be if and otherwise. Now, the series converges but since and the sequence does not converge.
3. The classic example is .
4. Since is decreasing it follows that . Let be the partial sum of the LHS. It follows that

The result now follows.

## Vanishing double summation

Let . Evaluate the series

Solution

Let and , then

since the sum over every triple  vanishes (one should also check that the sum absolutely converges but that’s straightforward by abelian summation).

The problem was first proposed on AoPS . There is a second solution though on MSE.

## A product

Let . Prove that:

Solution

The product eventually telescopes;

## Arctan integral

Prove that

where denotes the Euler – Mascheroni constant.

Solution

Beginning by parts we have,

However,

The result now follows taking and .

## Bessel function integral

Let denote the Bessel function of the first kind. Prove that

Solution

We recall that

Hence,

Then,

Using the fact that the looks like an ‘almost periodic’ function with decreasing amplitude. If we denote by the zeros of then as and furthermore

as for each . So the integral converges uniformly in this case justifying the interchange of limit and integral.

The result follows.