New rate of convergence of series and its application

Abstract

The existing rate of convergence of series requires a known limit of that series. Direct computation of Brun’s constant as the sum of twin prime reciprocals is challenging as the convergence is agonizingly slow. Here, I introduce the relative increase of the partial sum of a series as a new rate of convergence regardless of the limit of the series. With this concept, I extract a slowly convergent part from the original sum for Brun’s constant so that the remaining one converges faster. Computed data demonstrate the usefulness of the concept. As a result, I obtain an improved lower bound for Brun’s constant using less computing resource and establish an approximate mapping that gives a new estimate for the constant at 1.89558413.

Keywords

• Series
• rate of convergence
• Brun’s constant