Finding finite sums, products, and limits of some numerical sequences. Part 2. Application of methods of higher mathematics. Finding sums of series

Abstract

Compared to the methods of elementary mathematics, the methods of higher mathematics significantly expand the range of possibilities when finding sums and products of elements of some numerical sequences and sums of numerical or functional series. This article examines the application of de Moivre’s, Euler’s formulas and Newton's binomial, examples are given. Considering a suitable function on the segment $[a,b]$ and forming the corresponding integral Riemann sum for it, we find its value by integration. Using the Taylor & Maclaurin series of some known functions in $R$ or $C$, integrating and differentiating the corresponding power and functional series, we obtain the sums we are looking for. An example of the application of the Wallis formula is given. Expanding some functions into a Fourier series, we find for certain values ​​of the argument the sum of many interesting numerical series. An example of using Lyapunov's equality to calculate the sum of trigonometric series is given. The use of operational calculus methods for finding sums of numerical, functional and trigonometric series is considered, an example of the use of Dirac's $\delta $-function and its properties is given.