On a sum involving powers of reciprocals of an arithmetical progression

ON FINITE SUMS OF RECIPROCALS OF DISTINCT nTR POWERS

Introduction* It has long been known that every positive rational number can be represented as a finite sum of reciprocals of distinct positive integers (the first proof having been given by Leonardo Pisano [6] in 1202). It is the purpose of this paper to characterize {cf. Theorem 4) those rational numbers which can be written as finite sums of reciprocals of distinct nth. powers of integers, w...

Perfect powers in products of terms in an arithmetical progression III

Stable Laws for Sums of Reciprocals

We obtain the limiting behavior of the sum of reciprocal powers of a random sample. In particular, the mean sample reciprocal tends to a Cauchy distribution centered on the principal value (PV) of the mean population reciprocal. AMS 2000 subject classification: Primary 60E07; Secondary 60F05.

Sum-Rate Maximization Based on Power Constraints for Cooperative AF Relay Networks

In this paper, our objective is maximizing total sum-rate subject to power constraints on total relay transmit power or individual relay powers, for amplify-and-forward single-antenna relay-based wireless communication networks. We derive a closed-form solution for the total power constraint optimization problem and show that the individual relay power constraints optimization problem is a quad...

On the Sum of Reciprocals of Amicable Numbers

Two numbers m and n are considered amicable if the sum of their proper divisors, s(n) and s(m), satisfy s(n) = m and s(m) = n. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, P , is a constant. We obtain both a lower and an upper bound on the value of P .