Let be a sequence defined as
Find the supremum as well as the infimum of the sequence .
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,
Evaluate the series
and the problem is over.
Let be a compact metric space and let be a sequence of compact sets in . Is it true that is compact?
The answer is no since we can take and . Hence
which is clearly not compact.
Evaluate the limit
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.
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
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