Let such that . Evaluate the series:
Lemma 1: For the functions , it holds that:
Lemma 2: It holds that:
Proof: Simple calculations using Lemma reveal the identity.
Lemma 3: Using Lemma 2 it holds that
and as a consequence
Then, successively we have that:
Let . Prove that
It holds that
Note: Similarly, it holds that .
Find all functions such that and
First of all we note that
Since the integrand is positive it only remains that
Setting and at the last equation we have that:
Since is positive we conclude that
which satisfies the given conditions.
Let be a concave function. Prove that
Since is concave , it holds that
By setting and we get that . Thus,
and the exercise is complete.
Let be a sequence of positive real numbers such that
Prove that converges.
Fix and let . Then, there exist such that where . Thus,
Letting it follows that
Since this holds forall it follows that and the result follows.