## On the supremum and infimum of a sine sequence

Let be a sequence defined as

Find the supremum as well as the infimum of the sequence .

**Solution**

**Background: **This problem was on the shortlist of the 2014 Olimpiada Nationala de Matematica de Romania and was suggested by Leo Giugiuc.

We begin by the very well known manipulation.

Thus and we have to find the supremum and infimum of . Since the values are dense on the unit circle , the same shall hold for implying that and . Thus,

## An arctan series

Evaluate the series

*(Dan Sitaru)*

**Solution**

Well,

and the problem is over.

## Is the infinite union compact?

Let be a compact metric space and let be a sequence of compact sets in . Is it true that is compact?

**Solution**

The answer is no since we can take and . Hence

which is clearly not compact.

## A zeta tail limit

Evaluate the limit

**Solution**

One may use creative telescoping and deduce the double inequality

since . The limit follows to be .

Another way is using Riemann sum. Note that

*Choose which one you prefer the most. *

## A series with least common multiple

Let be a strictly increasing sequence of positive numbers. For all denote as the least common multiple of the first terms of the sequence. Prove that , as , the following sum converges

**Solution**

and the original question follows since the sum we seek is less or equal to .

However, we are presenting another proof. Denote as the average order of the numbers , i.e.,

For any we have where is the product of primes not present in the factorization of . Note that are squarefree integers. Note also that it may be an empty product, i.e., . Then

It is easy to see (and show by induction) that so we have

Hence, Consequently, we have

So the sum of reciprocals of converges. Then, by Cesàro summation, we see that

also converges.