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  2. Alternating series - Wikipedia

    en.wikipedia.org/wiki/Alternating_series

    Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.

  3. Alternating series test - Wikipedia

    en.wikipedia.org/wiki/Alternating_series_test

    In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.

  4. Integral test for convergence - Wikipedia

    en.wikipedia.org/wiki/Integral_test_for_convergence

    Once such a sequence is found, a similar question can be asked with f(n) taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series. Using the integral test for convergence, one can show (see below) that, for every natural number k, the series

  5. Direct comparison test - Wikipedia

    en.wikipedia.org/wiki/Direct_comparison_test

    In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.

  6. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]

  7. 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.

  8. Limit comparison test - Wikipedia

    en.wikipedia.org/wiki/Limit_comparison_test

    If diverges and converges, then necessarily =, that is, =. The essential content here is that in some sense the numbers a n {\displaystyle a_{n}} are larger than the numbers b n {\displaystyle b_{n}} .

  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