convergence test

(noun)

methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series

Related Terms

  • converge

Examples of convergence test in the following topics:

  • Tips for Testing Series

    • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
    • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
    • When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
    • Here is a summary for the convergence test that we have learned:
    • Formulate three techniques that will help when testing the convergence of a series
  • Comparison Tests

    • Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.
    • Comparison tests may mean either limit comparison tests or direct comparison tests.
    • The limit comparison test is a method of testing for the convergence of an infinite series, while the direct comparison test is a way of deducing the convergence or divergence of an infinite series or an improper integral by comparison with other series or integral whose convergence properties are already known.
    • The direct comparison test provides a way of deducing the convergence or divergence of an infinite series or an improper integral.
    • In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
  • Absolute Convergence and Ratio and Root Tests

    • The ratio test is a test (or "criterion") for the convergence of a series ∑n=1∞an\sum_{n=1}^\infty a_n∑​n=1​∞​​a​n​​, where each term is a real or complex number and ana_na​n​​ is nonzero when n is large.
    • if L=1L = 1L=1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
    • The root test is a criterion for the convergence (a convergence test) of an infinite series.
    • Note that if limn→∞∣an∣n\lim_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}lim​n→∞​​​n​​√​∣a​n​​∣​​​ converges, then it equals CCC and may be used in the root test instead.
    • otherwise the test is inconclusive (the series may diverge, converge absolutely, or converge conditionally).
  • The Integral Test and Estimates of Sums

    • The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
    • The integral test for convergence is a method used to test infinite series of non-negative terms for convergence.
    • Although we won't go into the details, the proof of the test also gives the lower and upper bounds:
    • In this way, it is possible to investigate the borderline between divergence and convergence of infinite series.
    • The integral test applied to the harmonic series.
  • Convergence of Series with Positive Terms

    • converge?
    • It is possible to "visualize" its convergence on the real number line?
    • For these specific examples, there are easy ways to check the convergence.
    • For these general cases, we can experiment with several well-known convergence tests (such as ratio test, integral test, etc.).
    • We will learn some of these tests in the following atoms.
  • Alternating Series

    • Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
    • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms ana_na​n​​ converge to 000 monotonically.
    • Therefore, our partial sum SmS_mS​m​​ converges.
    • an=1na_n = \frac1na​n​​=​n​​1​​ converges to 0 monotonically.
    • Therefore, the sum ∑n=1∞(−1)n+1n\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}∑​n=1​∞​​​n​​(−1)​n+1​​​​ converges by the alternating series test.
  • Summing an Infinite Series

    • Infinite sequences and series can either converge or diverge.
    • A series is said to converge when the sequence of partial sums has a finite limit.
    • By definition the series ∑n=0∞an\sum_{n=0}^\infty a_n∑​n=0​∞​​a​n​​ converges to a limit LLL if and only if the associated sequence of partial sums  converges to LLL.
    • An easy way that an infinite series can converge is if all the ana_{n}a​n​​ are zero for sufficiently large nnns.
    • This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy.
  • Power Series

    • A power series will converge for some values of the variable xxx and may diverge for others.
    • All power series f(x)f(x)f(x) in powers of (x−c)(x-c)(x−c) will converge at x=cx=cx=c.
    • If ccc is not the only convergent point, then there is always a number rrr with 0 < r ≤ ∞ such that the series converges whenever $\left| x-c \right| r$.
    • The number rrr is called the radius of convergence of the power series.
  • Series

    • By definition, the series ∑n=0∞an\sum_{n=0}^{\infty} a_n∑​n=0​∞​​a​n​​ converges to a limit LLL if and only if the associated sequence of partial sums {Sk}\{S_k\}{S​k​​} converges to LLL.
    • State the requirements for a series to converge to a limit
  • Sequences

    • The plot of a convergent sequence (ana_na​n​​) is shown in blue.
    • Visually, we can see that the sequence is converging to the limit of 000 as nnn increases.
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