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  2. Dirichlet's test - Wikipedia

    en.wikipedia.org/wiki/Dirichlet's_test

    An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals, and g is a non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.

  3. Limit comparison test - Wikipedia

    en.wikipedia.org/wiki/Limit_comparison_test

    In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]

  4. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.

  5. Ratio test - Wikipedia

    en.wikipedia.org/wiki/Ratio_test

    In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.

  6. General Dirichlet series - Wikipedia

    en.wikipedia.org/wiki/General_Dirichlet_series

    In the latter case, there exist a such that the series is convergent for > and divergent for <. By convention, σ c = ∞ {\displaystyle \sigma _{c}=\infty } if the series converges nowhere and σ c = − ∞ {\displaystyle \sigma _{c}=-\infty } if the series converges everywhere on the complex plane .

  7. nth-term test - Wikipedia

    en.wikipedia.org/wiki/Nth-term_test

    The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.

  8. Conditional convergence - Wikipedia

    en.wikipedia.org/wiki/Conditional_convergence

    A classic example is the alternating harmonic series given by + + = = +, which converges to ⁡ (), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem .

  9. Absolute convergence - Wikipedia

    en.wikipedia.org/wiki/Absolute_convergence

    If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.