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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.

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