Convergence of a series

An infinite series converges when its sequence of partial sums approaches a finite limit. That limit is the sum of the series.

Partial sum: A partial sum is the sum of the first finitely many terms of a sequence of terms. A series converges precisely when its partial sums approach a finite limit.
Infinite series: An infinite series is the formal sum of the terms of an infinite sequence. Convergence is a property of this infinite sum, defined through finite partial sums.
Divergence of a series: A series diverges when its partial sums fail to approach a finite limit. It is the direct alternative to convergence, including when partial sums grow or oscillate.
Power series: A power series is an infinite series whose terms are coefficients multiplied by powers of a variable. Its convergence determines where the represented function is defined by the expansion.
Cauchy criterion: A sequence is Cauchy if its terms become arbitrarily close to one another beyond some index. The partial sums converge exactly when their tails become arbitrarily small.
Sequence: A sequence is an ordered list of elements indexed by integers, usually natural numbers. The partial sums form a sequence whose limit defines series convergence.
Conditional convergence: A series converges conditionally when it converges but its series of absolute values diverges. Its convergence relies on cancellation among terms.
Taylor series: A Taylor series represents a function as a power series built from its derivatives at one point. Convergence alone does not ensure that the series equals the function.
Comparison test: A convergence test that compares nonnegative series term by term with a known series. It transfers convergence or divergence from a suitable benchmark series.
Limit of a sequence: A limit is a value that sequence terms approach as their indices increase. A series converges when its partial-sum sequence has a finite limit.

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