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# Tag Archives: Series

## Divergent Möbius series

Let denote the Möbius function. Prove that

Solution

Summing only over primes , where ,  we have that

## Application of extended binomial theorem

For the values of for which the following series makes sense, prove that

Solution

We have successively:

due to the extended binomial theorem.

## Double summation

Evaluate the sum

Solution

Lemma: It holds that

Proof: Consider the function and let us it integrate over the following contour

By the residue theorem it follows that

For the residues we have

The integrals along the sides vanish; hence:

and since

the result follows.

Back to the problem. We have successively:

## Arithmotheoretic sum

Evaluate the sum

Solution

The sum converges absolutely , so we can switch the order of summation; hence:

The last sum equals and hence

## Convergence of a series

Examine the convergence of a series

Solution

From Taylor’s theorem with integral remainder we have that

However it is known that . Hence,

Exponentiating we get

Thus the series converges.