## A beautiful limit with harmonic number

Let denote the – th harmonic number. Prove that

**Solution**

We note that

At the last integral we apply the substitution . Thus,

After all these we have that

proving the result.

## Zero function

Let such that and forall . Prove that forall .

**Solution**

We note that

Hence is decreasing. Therefore , . The result follows.

## 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,