Let denote the -th harmonic number. Prove that
Using the well known asymptotic formula for the – th harmonic number
we conclude that
We begin with the simple observation:
where is an integer. The last summand is of the form . Thus,
and the exercise comes to an end.
Let be a sequence of real numbers. Compute:
First and foremost we set and it is obvious that . We are making use of probabilistic methods. Suppose than an infinite number of coins are flipped. Let be the probability that the -th coin toss lands heads and let us consider the first time heads comes up. Then is the probability that the first head appears in the – th flip and is the probability that all flips come up tails. Thus,
Prove that in any triangle it holds that
where denotes the circumradius and the inradius.
Using the law of sines we have that
and if we denote the area of the triangle then
Let be three positive real numbers such that . Prove that
By AM – GM we have,
Hence and the exercise is complete.