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Evaluate the limit:
It is quite known that
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 .
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.
Let denote the Riemann zeta function. Evaluate the series:
Let . We are proving the more general result.
where denotes the digamma function.
First of all, we note that:
due to the reflection formula .
Side note: If then the sum equals .
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?
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,