Calculating the sum of a number series using a definite integral

Abstract

The theory of series is widely used for solving integrals, differential equations, and functions. In some cases, the sum of a
numerical series can be calculated, and sometimes it is better to use non-standard methods, as these approaches can significantly
reduce the time needed to solve them. The goal of this article is to establish integral equations to calculate the sum of certain
types of series, which can then be easily computed using these equations. After deriving the integral equations, several examples
are solved using the obtained equations.
The methods presented here strengthen the internal disciplinary connection between the theory of series and integral
calculus.
Knowledge of how to calculate such numerical series is essential for students of theoretical physics and technical
disciplines. Therefore, it is desirable to study such topics for them and provide examples to develop their problem-solving
abilities.