If ratio between next-term absolute value and previous-term absolute value is greater than or equal to one, sequence diverges {divergence, series}.
semiconvergence
Divergent series {semiconvergent series} can have nth-term error less than n+1th-term absolute value, so error decreases faster than terms increase. Though they diverge, semiconvergent series can evaluate integrals, because sum is finite. Useful asymptotic series is f(x) = a0 + a1/x + a2/(x^2) + ... If x is large, limit of x^n * (f(x) - series) = 0. x approaches 0 if 1/(x^n) * (f(x) - series) = a(n). Other ideas about asymptotic series include {Birkhoff's theorem} {Birkhoff's connection formula} {WKBJ solution} {Airy's integral}.
Mathematical Sciences>Calculus>Series>Convergence
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Date Modified: 2022.0224