## 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.

## The integral domain is a field

Prove that an integral domain with the property that every strictly decreasing chain of ideals must be finite in length is a field.

## Double inequality

In a triangle prove that

## Trigonometric series

Let . Prove that

**Solution**

First of all we note that

Hence,

Letting we get the requested value.