Infinite series

An infinite series is the limit, when it exists, of the sequence of finite partial sums of an ordered sequence of terms. It converges when those partial sums approach a finite value.

Partial sum: A finite sum of the first terms of a sequence, used to define an infinite series. The sequence of partial sums determines whether the series converges.
Sequence: An ordered list of elements indexed by natural numbers or another ordered set. A series is built from a sequence by adding its terms in order.
Conditional convergence: Convergence of a series whose series of absolute term values diverges. It contrasts with absolute convergence and makes rearrangement delicate.
Taylor series: A power series whose coefficients are determined by derivatives of a function at one point. It represents functions locally as infinite sums of polynomial terms.
Convergence test: A mathematical criterion for deciding whether a sequence or series converges. Tests establish convergence without requiring an explicit sum.
Limit of a sequence: A value that sequence terms approach as their indices increase without bound. An infinite series converges when its partial-sum sequence has a limit.
Divergent series: An infinite series whose sequence of partial sums has no finite limit. A term sequence tending to zero does not by itself ensure a sum exists.
Fourier series: An expansion of a periodic function into sine and cosine terms. Convergence of its partial sums governs approximation of periodic signals.
Cauchy criterion: A criterion stating that a sequence converges if and only if its terms eventually become arbitrarily close to one another. It characterizes series convergence by requiring sufficiently distant partial sums to be close.
Geometric series: A series whose successive terms have a constant ratio. Its convergence and sum follow directly from a finite partial-sum formula.

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