# Explicit Evaluations of Sums of Sequence Tails

## TL;DR

This paper employs Abel's summation formula to explicitly evaluate quadratic and cubic sums involving sequences and zeta functions, revealing new identities and relationships among harmonic numbers, multiple zeta values, and their variants.

## Contribution

It provides explicit formulas for sums of sequence tails using Abel's summation and derives new identities linking harmonic numbers to multiple zeta values.

## Key findings

- Explicit formulas for sums involving sequence tails and zeta functions.
- New identities connecting harmonic numbers with multiple zeta values.
- Evaluation of series involving multiple zeta star values.

## 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.