Generating formal series and applications

We deepen here the insight on formal power series. We temporarily abandon formality and consider the notion of the convergence of a power series; we’ll see in particular how a smart choice of a closed form of a given power series is useful to recover the sum of the power series. A large part of the chapter is devoted to determining the generating formal series for some notable sequences, including sequences of binomial coefficients, harmonic numbers, Stirling and Bell numbers, Eulerian numbers, as well as sequences of integral powers. One section is devoted to the Bernoulli numbers: not only do they allow us to express the sum of consecutive m-th powers of the natural numbers (Faulhaber’s formula), but they turn out to be useful, as we shall see in Chap. 13, in approximating the sum of the consecutive values on the natural numbers of any given smooth function. Some useful estimates of the Bernoulli numbers are given via the Riemann zeta function, namely the sum of the series of the inverses of a given real power of the natural numbers. Finally, a section is devoted to the applications of formal power series to probabilities.