Prove that the series
converges. Examine if the convergence is absolute.
Lemma: Let such that then the sequence
With the above lemma in mind the series converges as a consequence of Dirichlet’s theorem. The fact that the series does not converge absolutely can be seen by applying the Jordan inequality
hence the series diverges absolutely.
For what values of does the series
For none. Since for the sequence it holds that we deduce that the potential limits are or . Hence, the sequence cannot tend to zero and the conclusion follows.
Note: One interesting question is the following. For what values of does the series
converge? The answer may be found at this link.
Find all positive integers such that it holds that
where stands for the period.
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,