An infinite series is an expression of the form = a1 + a2 + ... + an + ..., in which an infinite number of terms (complex and real numbers) are added. The sum of the first n terms of a series Sn = is a partial sum of a series. If there is a finite limit S of partial sums for n → ∞, then it is the sum of a series and the series is convergent (S = limn→∞Sn); otherwise, the series is called divergent. The series is absolutely convergent if a series converges.
If the terms of a series are presented by the functions of some variable, i.e., ak = ak(x) then the series is called functional.
A series of functions (a1(x) + ... ak(x)) converges to a function (a sum) S(x) for each value of x, which has a finite limit
A series converges uniformly on a set of values x [a, b] to S(x) if a sequence of its partial sums Sn converges uniformly, i.e., for ε > 0 N independent of x [a, b], such that the condition |S(x)| − |Sn(x)| < ε is met for all n ≥ N.
The most widely used criterion for uniform convergence is the Weierstrass M-test: if the terms of a functional series (x) on the interval [a, b] satisfy the inequality |ak(x)| ≤ Mk, where Mk are the terms of some convergent series , then the series converges on [a, b] uniformly and absolutely.
To define uniform convergence of series of the form bk(x), Abel's test for uniform convergence is used: if a series converges uniformly on the interval [a, b], and a function ak(x) forms a monotone bounded for any x and k sequence (i.e., |ak(x)| ≤ M), then the series bk(x) converges uniformly on [a, b].
Uniformly-convergent series can be added term by term and multiplied by a bounded function. The series obtained as a result are also convergent. If separate terms of a uniformly-convergent series are continuous functions, then the sum of a series is also a continuous function. If the function ak(x) are integrable on the interval [a, b], a series consisting of the integrals converges uniformly; then this series can be integrated termwise, with the sum of the integrals being equal to the integral of the sum
If a function ak(x) has continuous derivatives and the series composed of derivatives uniformly converges, then the series ak(x) can be differentiated term by term, with S'(x) = .
The expansion of functions into infinite series is often employed in solving problems of hydrodynamics and heat transfer. With the help of series, the exact values of transcendental constants, functions and integrals are calculated and differential equations are solved.