## Limit with harmonic number

Let denote the -th harmonic number. Prove that

**Solution**

Using the well known asymptotic formula for the – th harmonic number

we conclude that

## Limit of

Prove that

**Solution**

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.

## On an infinite summation

Let be a sequence of real numbers. Compute:

**Solution**

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,

## Trigonometric equality

Prove that in any triangle it holds that

where denotes the circumradius and the inradius.

**Solution**

Using the law of sines we have that

and if we denote the area of the triangle then

Thus,

## Root inequality

Let be three positive real numbers such that . Prove that

**Solution**

By AM – GM we have,

However,

Hence and the exercise is complete.