Convergent series

An infinite series is convergent when its sequence of partial sums approaches a finite limit. Its sum is that limit.

Sequence of partial sums: The sequence formed by adding the first n terms of a series for each positive integer n. A series converges precisely when this sequence approaches a finite limit.
Infinite series: A sum of the terms of an infinite sequence, defined through the limit of its partial sums. Convergent series are the infinite series whose partial sums have finite limits.
Divergent series: An infinite series whose sequence of partial sums does not approach a finite limit. It is the direct alternative: a series either converges or diverges.
Taylor series: A power series whose coefficients are determined by the derivatives of a function at a point. Where it converges to the function, it represents that function through an infinite sum.
Cauchy criterion: A sequence converges in the real numbers exactly when its terms eventually become arbitrarily close to one another. It tests convergence through the tails of partial sums without requiring their limit.
Sequence: An ordered list of objects, often numbers, indexed by positive integers. The terms of a series form a sequence whose partial sums determine convergence.
Term test for divergence: The result that a series diverges if its terms fail to approach zero. Terms tending to zero are necessary for convergence, but do not guarantee it.
Power series: A series of powers of a variable multiplied by fixed coefficients. Its convergence often holds inside a radius that determines where the representation is valid.
Ratio test: A test that compares successive term magnitudes to determine whether a series converges absolutely or diverges. It gives a quick verdict when successive terms have a stable ratio.
Limit of a sequence: A value that sequence terms approach as their indices increase without bound. This is the limiting behavior required of a convergent series' partial sums.

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