Abel's test

Abel's test is a criterion for convergence of a series of products: it applies when one factor has convergent partial sums and the other is monotone and bounded.

Summation by parts: A finite-sum identity that transfers differences between two sequences, analogous to integration by parts. It converts the product series into a sum controlled by partial sums of the factor that converges.
Infinite series: A sum defined as the limit of the partial sums of an infinite sequence of terms. Abel's test decides whether a particular infinite series has a finite sum.
Power series: A series whose terms are coefficients multiplied by nonnegative integer powers of a variable. Abel's theorem uses endpoint convergence to control a power series throughout its interval of convergence.
Niels Henrik Abel: A Norwegian mathematician whose work shaped the theory of elliptic functions, algebra, and infinite series. The convergence test bears Abel's name, though it is distinct from Abel's theorem.
Cauchy criterion: A criterion stating that a sequence converges exactly when its terms eventually become arbitrarily close to one another. Abel's test proves convergence by showing that every sufficiently late tail of the series is small.
Convergence of a series: Convergence of a series means that its sequence of partial sums approaches a finite limit. This is the precise outcome Abel's test establishes.
Abel's theorem: A theorem relating the limit of a power series at a boundary point to the sum of its coefficient series. It is a major power-series application of the same partial-sum and monotonicity principle.
Abel's transformation: A discrete summation identity expressing a weighted sum through partial sums and successive differences of weights. This transformation supplies the algebraic step behind Abel's test.
Partial sum: The sum of the first finitely many terms of a sequence or series. Convergence of the partial sums of one factor is the test's central hypothesis.
Absolute convergence: A series converges absolutely when the series of absolute values of its terms converges. Abel's test guarantees ordinary convergence, which need not be absolute.

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Conditional convergenceRelated: It extends convergence arguments to weighted series that may not converge absolutely.