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