Some general summation results with applications to arctangent-type series

Abstract

This article extends the telescopic summation and Abel summation by parts methods by introducing various tuning parameters and examining sequences of variable-dependent functions. In a substantial part, these general results are applied to arctangent-type series. We re-examine known results from a fresh viewpoint and establish numerous new ones, with an emphasis on establishing unified formulas applicable in various scenarios. In particular, some new arctangent-type series defined with the binomial coefficients and others with the Fibonacci sequence are demonstrated. Therefore, this work contributes to the advancement of telescopic techniques and enriches our understanding of arctangent-type series.