Abel's inequality

Abel's inequality bounds a weighted sum using bounds on the partial sums of one sequence and monotonicity of the weights. It follows from the discrete summation-by-parts identity.

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Summation by parts: A discrete identity that rewrites a sum of products using partial sums and successive differences. The identity converts the weighted sum into terms directly controlled by Abel's hypotheses.
Finite sequence: An ordered list of a fixed, finite number of terms. Abel's inequality is first stated for sums over a finite index range.
Dirichlet test: A convergence test for series whose partial sums are bounded and whose other factor decreases to zero. Its proof applies the same partial-sum and monotone-weight estimate.
Abel's lemma: A summation-by-parts identity expressing a weighted sum through partial sums and differences of weights. It is the algebraic lemma behind the bound, rather than the bound itself.
Partial sum: A sum of consecutive terms from the beginning of a sequence through a specified index. Bounds on these cumulative sums supply the sequence-side estimate.
Bounded sequence: A sequence whose terms all have absolute value below a common finite bound. A common version assumes the partial sums stay uniformly bounded.
Convergence of series: The property that the partial sums of an infinite series approach a finite limit. Abel's inequality provides estimates used to establish series convergence.
Abel's theorem: A theorem relating the boundary convergence of a power series to its behavior inside the unit disk. Its similar name masks a distinct result about power-series limits.
Monotone sequence: A sequence whose terms are either nondecreasing throughout or nonincreasing throughout. Monotonicity fixes the signs of successive weight differences in the summation-by-parts formula.
Nonnegative sequence: A sequence whose terms are all greater than or equal to zero. Nonnegative weights make the direction and size of the bound especially transparent.