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## A series limit

Evaluate the limit:

**Solution**

It is quite known that

Thus,

Therefore, by induction , for ,

Summing the last equation we get:

and raising the last equation to the power , we get:

Thus, by the squeeze theorem the limit is equal to .

## A limit on Dirichlet function

Let be the Dirichlet function;

Evaluate the limit

We simply note that

and the limit follows to be . The * reason *why

is because is rational if-f is a perfect square.

## Series of zeta sum

Let denote the Riemann zeta function. Evaluate the series:

**Solution**

Let . We are proving the more general result.

where denotes the digamma function.

First of all, we note that:

Thus,

due to the reflection formula .

**Side note: **If then the sum equals .

## Lebesgue measure of Cantor set

An alternative way to define the Cantor set is the following:

What is the Lebesgue measure of the Cantor set if we consider it as a subset of ? Is countable?

## A divergent series …. or maybe not?

The number ranges over all possible powers with both the base and the exponent positive integers greater than , assuming each such value only once. Prove that:

Let us denote by the set of positive integers greater than that are not perfect powers ( i.e are not of the form , where is a positive integer and ). Since the terms of the series are positive , we can freely permute them. Thus,