Absolute convergence

A series or integral converges when its terms or integrand are replaced by their absolute values. For series, this guarantees convergence of the original series regardless of term order.

Comparison test: A convergence test that compares nonnegative terms with those of a known series or integral. It proves absolute convergence by bounding absolute terms with a convergent comparison.
Rearrangement theorem: The theorem that rearranging an absolutely convergent series preserves its sum. Absolute convergence prevents term reordering from changing the series' value.
Conditional convergence: Convergence of a series whose series of absolute values diverges. It separates convergence that survives absolute values from convergence that does not.
Infinite series: A sum defined as the limit of the partial sums of a sequence of terms. Absolute convergence is a strengthened convergence condition for infinite series.
Ratio test: A series convergence test based on the limiting ratio of successive term magnitudes. It detects absolute convergence when successive absolute terms eventually shrink geometrically.
Fubini's theorem: A theorem that permits changing the order of integration under suitable integrability conditions. Absolute integrability supplies a central condition for safely exchanging integration order.
Alternating series test: A test guaranteeing convergence for alternating terms that decrease in magnitude to zero. It can establish convergence even when the absolute-value series diverges.
Partial sum: The finite sum of the first specified number of terms in a sequence or series. A series converges only if its partial sums approach a finite limit.
Root test: A series convergence test based on the limiting nth root of each term's magnitude. It identifies absolute convergence when the nth-root limit is below one.
Power series: An infinite series whose terms are coefficients multiplied by powers of a variable. Inside its radius of convergence, a power series converges absolutely.

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