On the flint hills series

TL;DR

This paper investigates the convergence of a specific infinite series involving sine functions by relating it to other series through a novel method involving constructing pillars near series terms.

Contribution

It introduces a new approach to analyze series convergence by connecting it to more manageable series using geometric constructions.

Findings

The convergence depends on a complex inequality involving binomial sums and sine functions.

The method relates series convergence to inequalities that involve sine squared and polynomial growth.

The analysis provides conditions under which the series converges or diverges.

Abstract

In this note, we study the flint hills series of the form \begin{align} \sum \limits_{n=1}^{\infty}\frac{1}{(\sin^2n) n^3}\nonumber \end{align} via a certain method. The method essentially works by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular, we show that the convergence of the flint hill series relies very heavily on the condition that for any small $\epsilon>0$ \begin{align} \bigg|\sum \limits_{i=0}^{\frac{n+1}{2}}\sum \limits_{j=0}^{i}(-1)^{i-j}\binom{n}{2i+1} \binom{i}{j}\bigg|^{2s} \leq |(\sin^2n)|n^{2s+2-\epsilon}\nonumber \end{align} for some $s\in \mathbb{N}$.