Some Special Series and Their Applications

Abstract

In this chapter we assemble a few results about two special types of series, namely,

$$ \frac{1}{2} {{a}_{0}} + \sum\limits_{{n = 1}}^{\infty } {{{a}_{n}}\cos nx = \sum\limits_{{n \in z}} {{{c}_{n}}{{e}^{{inx}}},} } $$

where ${{c}_{n}} = \frac{1}{2} {{a}_{{\left| n \right|}}};$ and

$$\sum\limits_{n = 1}^\infty {{a_n}\sin nx = \sum\limits_{n \in z} {{c_n}{e^{inx}},} } $$

where c_π = (l/2i) sgn n • a_|n|. We shall assume throughout that the a_n are real-valued, and write s_N and σ_N for the Nth partial sum and the Nth Cesàro mean, respectively, of whichever series happens to be under discussion.