4. Sums and Series

In this chapter, we experience the problems of the second major area of the book, which is Sums and Series. Like in the first chapter, the reader will encounter a wide palette of irresistible problems, and some of the sections will also include challenging questions. As examples, we mention interesting and useful finite sums involving the central binomial coefficient; series with the central binomial coefficient; lots of (less known or not known) Cauchy products; generating functions involving generalized harmonic numbers and generalized skew-harmonic numbers; atypical harmonic series of weights 4 and 5; generalized atypical harmonic series; two (very) nice sets of atypical harmonic series involving the tail of the Riemann zeta function; the practical Fourier series expansions of the Bernoulli polynomials; sets of powerful good-to-know Fourier series of functions involving logarithms, polylogarithms, and trigonometric functions; marvelous binoharmonic series; and very useful polylogarithmic values involving a complex arguments.