The geometric series formula and its applications

TL;DR

This paper explores the geometric series formula using Lambert W function, deriving new limits of convergence and expressing the harmonic series in terms of special functions, confirming its divergence.

Contribution

It introduces a novel expression for the convergence limit of geometric series and relates the harmonic series to Lambert W function, providing new analytical insights.

Findings

Derived a new limit formula for geometric series convergence

Expressed the harmonic series using Lambert W function

Confirmed the divergence of the harmonic series through new expressions

Abstract

Let $n$ be an integer and $W_n$ be the Lambert $W$ function. Let $\log$ denote the natural logarithm so that $\delta=-W_n(-\log2)/\log2$. Given that $a$ and $r$ are respectively the first term and the constant ratio of an infinite geometric series, it is proved that the limit of convergence of the geometric series is $\displaystyle\lim_{n\to\pm\infty}{a\big[r^\delta-1\big]\big[r-1\big]^{-1}}$ where $r\neq1$. By applying the geometric series formula above, it is further proved that the harmonic series $\zeta(1)$ is given by $\zeta(1)=-2\big[\log2+W_n(-\log2)\big]$ and as $n\rightarrow\pm\infty$, the value of $\zeta(1)$ grows very slowly toward $\tilde\infty$, confirming the divergence of the harmonic series.