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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.

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