convergence test

(noun)

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

Related Terms

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 $\sum_{n=1}^\infty a_n$, where each term is a real or complex number and $a_n$ is nonzero when n is large.
    • if $L = 1$ 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 $\lim_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$ converges, then it equals $C$ 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 $a_n$ converge to $0$ monotonically.
    • Therefore, our partial sum $S_m$ converges.
    • $a_n = \frac1n$ converges to 0 monotonically.
    • Therefore, the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ 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 $\sum_{n=0}^\infty a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums  converges to $L$.
    • An easy way that an infinite series can converge is if all the $a_{n}$ are zero for sufficiently large $n$s.
    • This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy.

  • Power Series

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

  • Series

    • By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$.
    • State the requirements for a series to converge to a limit

  • Sequences

    • The plot of a convergent sequence ($a_n$) is shown in blue.
    • Visually, we can see that the sequence is converging to the limit of $0$ as $n$ increases.