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# Tag Archives: Series

## An series related to the eta Dedekind function

Prove that

**Solution**

Let us consider the function

and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,

We note that and that .

It’s also easy to see that

Hence, as we have that

By Residue theorem we have that

It is straightforward to show that

Hence,

in the limit . The result follows.

## Convergence of series

Let denote the greatest prime factor of . For example , . Define . Examine if the sum

converges.

**Solution**

Let be the -th prime number.

If , then with , and . It folows that

From Merten’s theorem

and the original series has the same character as

which is convergent.

## On a prime summation

Let denote the – th prime. Evaluate the sum

## Square summable

Let be a real sequence such that

If is a real sequence that is square summable; i.e the sequence converges.

Prove that is also square summable.

**Solution**

Let be defined as

where . We note that

Equality holds when . Hence, . From the hypothesis, it follows that is pointwise bounded. It follows from the Uniform boundedness principle ( Banach – Steinhaus ) that are bounded. Hence, is square summable.

## Trigonometric series

Let . Prove that

**Solution**

First of all we note that

Hence,

Letting we get the requested value.