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  2. Harmonic series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Harmonic_series_(mathematics)

    The series = + = + + is known as the alternating harmonic series. It is conditionally convergent by the alternating series test , but not absolutely convergent . Its sum is the natural logarithm of 2 .

  3. Alternating series test - Wikipedia

    en.wikipedia.org/wiki/Alternating_series_test

    The alternating series test guarantees that an alternating series ... the partial sum of the harmonic series, which is divergent. Hence the original series is ...

  4. 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.

  5. Riemann series theorem - Wikipedia

    en.wikipedia.org/wiki/Riemann_series_theorem

    In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.

  6. Convergent series - Wikipedia

    en.wikipedia.org/wiki/Convergent_series

    The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true.

  7. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    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]

  8. 1 + 2 + 3 + 4 + ⋯ - ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E...

    Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. [6] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below).

  9. nth-term test - Wikipedia

    en.wikipedia.org/wiki/Nth-term_test

    Many authors do not name this test or give it a shorter name. [2] When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.