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Sequences and Series - Convergence Tests

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Geometric series test

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Converges if the absolute value of the common ratio r<1|r| < 1; diverges otherwise.

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Limit comparison test

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If limnanbn=c\lim_{{n \to \infty}} \frac{a_n}{b_n} = c where cc is a positive finite number and bn\sum b_n converges, then an\sum a_n also converges; if bn\sum b_n diverges, then an\sum a_n diverges.

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Gregory's test

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A particular form of the alternating series test, where convergence is determined if the alternating sum of the sequence's terms tends to a limit.

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Dirichlet's test

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Converges if ana_n is a monotonic decreasing sequence that tends to 00 and bnb_n is bounded, with partial sums also bounded.

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Comparison test

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If 0anbn0 \leq a_n \leq b_n and bn\sum b_n converges, then an\sum a_n also converges; if an\sum a_n diverges, then bn\sum b_n also diverges.

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Bertrand's test

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Converges if limnn(log(n)(anan+11)1)>0\lim_{{n \to \infty}} n(\log(n)\left(\frac{a_n}{a_{n+1}} - 1\right) - 1) > 0; otherwise, diverges.

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Integral test

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If f(n)=anf(n) = a_n, the series converges if 1f(x)dx\int_{1}^{\infty} f(x) dx converges; diverges otherwise.

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Logarithmic test

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Converges if limnloganlogn>1\lim_{{n \to \infty}} \frac{\log |a_n|}{\log n} > 1; otherwise, diverges.

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Kummer's test

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Converges if there is a positive sequence cnc_n such that limncn(anan+11)>1\lim_{{n \to \infty}} c_n(\frac{a_n}{a_{n+1}} - 1) > 1; otherwise, diverges.

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Lambert's test

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Converges if nan0na_n \rightarrow 0 as nn \rightarrow \infty for an>0a_n > 0 and anan+1a_n \geq a_{n+1} for all nn.

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Stirling's test

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Converges if there is a constant C>0C > 0 such that anan+1=1+Cn+O(1n2)\frac{a_n}{a_{n+1}} = 1 + \frac{C}{n} + O(\frac{1}{n^2}).

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P-series test

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Converges if p>1p > 1; diverges for p1p \leq 1.

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Riemann series theorem

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Any conditionally convergent series can be rearranged to converge to any number or even diverge.

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Silverman's game

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Illustrates a series's convergence via a game with prescribed moves leading to an ever-decreasing remainder; not a formal test.

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Euler's convergence test

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Converges when the product of nn times the difference between consecutive terms ana_n and an+1a_{n+1} is convergent.

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Absolute convergence test

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A series converges absolutely if the sum of the absolute values of its terms converges.

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Cauchy's root test for convergence

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Equivalent to the root test; converges if limnan1n<1\lim_{{n \to \infty}} |a_n|^\frac{1}{n} < 1; otherwise, diverges or is inconclusive.

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Tauber's test

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Converges if nanna_n is bounded and ana_n tends to 00 as nn tends to infinity.

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Ratio test

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Converges absolutely if limnan+1an<1\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| < 1; diverges if the limit is >1> 1 or infinity; test is inconclusive if the limit equals 11.

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Cauchy convergence criterion

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A sequence converges if for all ϵ>0\epsilon > 0, there exists an NN such that for all m,nNm, n \geq N, aman<ϵ|a_m - a_n| < \epsilon.

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Borel's criterion

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A series converges if for all ϵ>0\epsilon > 0, there exist only finitely many natural numbers nn such that nan>ϵna_n > \epsilon.

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Gauss's test

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Converges if there is a positive number MM and p>1p > 1 such that limnnp(anan+11)>M\lim_{{n \to \infty}} n^p\left(\frac{a_n}{a_{n+1}} - 1\right) > M; otherwise, diverges.

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Frobenius method

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Used primarily for power series solutions to differential equations; convergence depends on the radius of convergence around singular points.

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Root test

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Converges absolutely if limnann<1\lim_{{n\to\infty}} \sqrt[n]{|a_n|} < 1; diverges if the limit is >1> 1 or infinity; test is inconclusive if the limit equals 11.

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Conditional convergence test

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A series is conditionally convergent if it converges but does not converge absolutely.

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Alternating series test

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Converges if the sequence ana_n is decreasing and limnan=0\lim_{{n \to \infty}} a_n = 0.

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D'Alembert's ratio test for convergence

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Equivalent to the ratio test; converges if limnan+1an<1\lim_{{n \to \infty}} \frac{|a_{n+1}|}{|a_n|} < 1; otherwise, diverges or is inconclusive.

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Leibniz's test for alternating series

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Identical to the alternating series test; converges if the absolute value of the terms decreases monotonically to 00.

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Gauss's test for convergence

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Similar to Gauss's test but modified specifically for hypergeometric series; involves complex analysis for determination.

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Hadamard's formula

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Used to find the radius of convergence RR of a power series; R=1lim supnannR = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}.

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The area convergence test

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For an alternating series where terms are represented by the areas of successive vertical rectangles under a decreasing function, the series converges.

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Raabe's test

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Converges if limnn(anan+11)>1\lim_{{n \to \infty}} n\left(\frac{a_n}{a_{n+1}} - 1\right) > 1; diverges if it is <1< 1; inconclusive if it equals 11.

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Abel's test

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Converges if ana_n is a monotonic convergent sequence and bnb_n is a bounded sequence.

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