Integrationsreihen

Integrationsreihen are a mathematical concept used to describe the representation and evaluation of integrals via infinite series. In analysis, one often expresses a function f on a closed interval [a,b] as a series f(x) = ∑_{n=0}^∞ f_n(x) and then integrates term by term, under suitable convergence conditions.

Definition and key idea: If the series ∑ f_n converges to f uniformly on [a,b], and the series of integrals ∑ ∫_a^b f_n(x) dx converges, then the definite integral of f can be computed as ∫_a^b f(x) dx = ∑_{n=0}^∞ ∫_a^b f_n(x) dx. Uniform or absolute convergence, or the use of convergence theorems (such as the dominated convergence theorem), are usually required to justify interchanging summation and integration.

Construction and examples: A common approach is to expand the integrand into a known series and then integrate termwise. For instance, expanding 1/(1+x) as a geometric series ∑_{n=0}^∞ (-1)^n x^n for |x|<1 allows termwise integration on any subinterval of (-1,1). Another example is expanding e^x as ∑_{n=0}^∞ x^n/n!, and integrating on [0,1] to obtain ∑_{n=0}^∞ 1/((n+1)n!).

Convergence and limitations: The legitimacy of termwise integration depends on the type of convergence and the interval considered. If the series does not converge uniformly, or if integrals diverge, interchanging summation and integration is not valid.

Applications: Integrationsreihen are used to evaluate definite integrals, to derive moments in probability, and to obtain analytical approximations in physics and engineering where integrals are difficult to compute directly. They connect to related methods such as power series integration, Fourier series, and Laplace transforms.