If successive-term absolute value is less than previous-term absolute value, series converges {absolutely convergent} {convergence, series}.
Series can be convergent even if successive-term absolute value is not less than previous-term absolute value {conditionally convergent}. Conditionally convergent series can rearrange to make sum be any number.
constant times sequence
If sequence converges, limit of constant times sequence is constant times sequence limit.
uniform
Absolute value of partial sum S(n) minus sum from x = 1 to x = n of S(n) * x can be less than small value, for all x {uniform convergence}.
Convergent-power-series partial-sum limit is partial sum, if convergence radius replaces general-term independent variable {Abel summability}.
Divergent series {asymptotic series} can represent functions and evaluate integrals. In such divergent series, nth-term error is less than n+1th-term absolute value, so error becomes less as term number increases.
Absolute value of each sequence term can be less than or equal to a constant {bounded sequence}.
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}.
If sequence successive term is less than previous term, and if general term is always less than constant, sequence has limit {limit, sequence} {sequence limit} {limit, series} less than or equal to constant.
Summable series can make convergent series {Tauberian theorem}.
Independent variable can have value {radius of convergence} {convergence radius} greater than zero at which power series changes from convergence to divergence. Power series, power-series differential, and power-series integral have same convergence radius.
For complex-number power series, if complex number lies within a complex-plane circle {convergence circle} {circle of convergence} centered on zero, with no singularities, series converges. If complex number lies outside a complex-plane circle, series diverges.
Laurent series has complex number that lies within annulus in complex plane {convergence annulus} {annulus of convergence}.
Independent variable can have values {region of convergence} {convergence region, series} for which series converges.
If and only if absolute value of difference between successive partial sums is less than small value {Cauchy convergence criterion}, series converges.
Sequence general term can be less than or equal to constant times second-sequence general term {comparison test}. If second sequence diverges, first sequence diverges. If second sequence converges, first sequence converges.
If second sequence converges, and if second-sequence general term divided by sequence general term has limit, first sequence converges.
If second sequence diverges, and second-sequence general term divided by sequence general term has limit or if quotient is infinite, first sequence diverges.
If limit of quotient of sequence general terms does not equal zero, both sequences either diverge or converge.
Integral from x = 0 to x = a of (f(x) * sin(u*x) / sin(x)) * dx, and integral from x = a to x = b of (f(x) * sin(u*x) / sin(x)) * dx {Dirichlet integral}, where b > a > 0, can show convergence.
Sequences converge if and only if integral of general term, from x equals some value to x equals infinity, exists {integral test}.
Alternating sequences can converge {Leibniz's test} {Leibniz test}.
If successive-term to previous-term ratio limit is less than one, sequence converges {ratio test}. If successive-term to previous-term ratio limit is greater than one, sequence diverges. If successive-term to previous-term ratio limit is one, sequence can converge or diverge. If general-term limit equals zero, successive-term to previous-term-ratio absolute-value limit is less than one. Generalized ratio test {d'Alembert's test} exists.
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Date Modified: 2022.0225