Some bounds for alternating Mathieu type series

New upper bounds for Mathieu–type series

The Mathieu’s series S(r) was considered firstly by É.L. Mathieu in 1890; its alternating variant S̃(r) has been recently introduced by Pogány et al. [12] where various bounds have been established for S, S̃. In this note we obtain new upper bounds over S(r), S̃(r) with the help of Hardy–Hilbert double integral inequality. 2000 Mathematics Subject Classification. Primary: 26D15, 33E20.

Some families of Mathieu a-series and alternating Mathieu a-series

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu a-series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu a-series. Relationships of the Mathieu a-series with the Ri...

Some New Bounds for Mathieu’s Series

The Cesàro Sums of Some Alternating Series

We compute the Cesàro sum of the series P n≥0(−1) P (x + n) where P (x) is a real polynomial. 1. THE MAIN PROBLEM Before we can formulate the main problem of this paper we need to recall of few classic facts. Denote by N the set of nonnegative integers. We will denote by R[[t]] the ring of formal power series with coefficients in R and by SeqR the vector space of sequences (an)n≥0 in R. We will...

On Some Properties of the Lüroth-type Alternating Series Representations for Real Numbers

We investigate some properties connected with the alternating Lüroth-type series representations for real numbers, in terms of the integer digits involved. In particular, we establish the analogous concept of the asymptotic density and the distribution of the maximum of the first n denominators, by applying appropriate limit theorems. 2000 Mathematics Subject Classification. 60A10, 11K55, 37A30...