numericseries

Numeric series, or simply a series, is the sum of the terms of a sequence. If {a_n} is a sequence, the series ∑_{n=1}^∞ a_n is defined as the limit, if it exists, of the sequence of partial sums S_N = ∑_{n=1}^N a_n. The study of series concerns whether this limit exists and, if so, its value, as well as the properties of the terms that influence convergence.

A series is convergent if the partial sums approach a finite number L as N increases. If no finite limit exists, the series is divergent. A convergent series may be absolutely convergent (∑ |a_n| converges) or conditionally convergent (∑ a_n converges but ∑ |a_n| diverges).

Convergence is analyzed using tests such as the geometric series test (|r|<1), the comparison test, the ratio test, the root test, and the alternating series test. Additional structure occurs in special types such as power series ∑ c_n x^n, where convergence is studied in terms of x and often yields a radius of convergence.

Examples: Arithmetic series have terms in an arithmetic progression; the partial sum is S_n = n/2 (a_1 + a_n). Geometric series ∑ a r^{n-1} has sum S = a/(1−r) when |r|<1. The harmonic series ∑ 1/n diverges; the alternating harmonic series ∑ (-1)^{n+1}/n converges (to ln 2).

Beyond pure theory, series appear in numerical methods, power series representations of functions, Fourier and other expansions, and in solving differential equations, among many areas. The concept of a numeric series is closely related to convergence, limits, and approximation.