Conditional convergence

A series converges to a finite sum while the series of absolute values of its terms diverges. Its convergence depends on cancellation between positive and negative terms.

Alternating series test: A test stating that an alternating series converges when term magnitudes decrease to zero. It gives a common sufficient condition for conditional convergence.
Infinite series: The limit of the partial sums of a sequence of terms, when that limit exists. Conditional convergence is a particular way an infinite series can converge.
Riemann series theorem: A theorem that a conditionally convergent real series can be rearranged to converge to any prescribed real number or diverge. It shows how strongly order can affect conditionally convergent sums.
Alternating p-series: The series with terms alternating in sign and magnitudes equal to reciprocal powers of positive integers. For exponents between zero and one, it converges conditionally.
Leibniz test: The alternating series test, named for Gottfried Wilhelm Leibniz. This classical form establishes convergence for alternating harmonic terms.
Absolute convergence: Convergence of a series of terms' absolute values. It is the stronger condition that conditional convergence explicitly fails.
Rearrangement of series: Changing the order of terms in an infinite series. Unlike absolute convergence, conditional convergence does not generally preserve the sum under rearrangement.
Dirichlet eta function: The function defined by the alternating Dirichlet series, initially where that series converges. Its defining series is conditionally convergent for real inputs between zero and one.
Harmonic series: The series whose terms are the reciprocals of the positive integers; it diverges. Its positive terms show why cancellation, rather than small terms alone, matters.
Absolute value: A real or complex number's nonnegative magnitude, disregarding its sign or direction. Taking each term's magnitude removes the cancellation that can enable conditional convergence.

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