Explicit Evaluations of Sums of Sequence Tails

Abstract

In this paper, we use Abel's summation formula to evaluate several quadratic and cubic sums of the form: \[{F_N}\left( {A,B;x} \right) := \sum\limits_{n = 1}^N {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right){x^n}} ,\;x \in [ - 1,1]\] and \[F\left( {A,B,\zeta (r)} \right): = \sum\limits_{n = 1}^\infty {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right)\left( {\zeta \left( r \right) - {\zeta_n}\left( r \right)} \right)} ,\] where the sequences $A_n,B_n$ are defined by the finite sums ${A_n} := \sum\limits_{k = 1}^n {{a_k}} ,\ {B_n} := \sum\limits_{k = 1}^n {{b_k}}\ ( {a_k},{b_k} =o(n^{-p}),{\mathop{\Re}\nolimits} \left( p \right) > 1 $) and $A = \mathop {\lim }\limits_{n \to \infty } {A_n},B = \mathop {\lim }\limits_{n \to \infty } {B_n},F\left( {A,B;x} \right) = \mathop {\lim }\limits_{n \to \infty } {F_n}\left( {A,B;x} \right)$. Namely, the sequences $A_n$ and $B_n$ are the partial sums of the convergent series $A$ and $B$, respectively. We give an explicit formula of ${F_n}\left( {A,B;x} \right)$ by using the method of Abel's summation formula. Then we use apply it to obtain a family of identities relating harmonic numbers to multiple zeta values. Furthermore, we also evaluate several other series involving multiple zeta star values. Some interesting (known or new) consequences and illustrative examples are considered.