Examples of convergence test in the following topics:
-
- 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 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.
-
- 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 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.
-
- 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.
-
- 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.
-
- The $t$-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances.
- The Welch's $t$-test is a nearly exact test for the case where the data are normal but the variances may differ.
- Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
- If $X_n$ converges in distribution to a random element $X$, and $Y$ converges in probability to a constant $c$, then:
- The nonparametric counterpart to the paired samples $t$-test is the Wilcoxon signed-rank test for paired samples.
-
- Letting "test" represent a parallel form of the test, the symbol rtest,test is used to denote the reliability of the test.
- Construct validity can be established by showing a test has both convergent and divergent validity.
- A test has convergent validity if it correlates with other tests that are also measures of the construct in question.
- Of course, some constructs may overlap so the establishment of convergent and divergent validity can be complex.
- Convergent and divergent validity could be established by showing the test correlates relatively highly with other measures of spatial ability but less highly with tests of verbal ability or social intelligence.
-
- It can be used as an alternative to the paired Student's $t$-test, $t$-test for matched pairs, or the $t$-test for dependent samples when the population cannot be assumed to be normally distributed.
- The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples.
- In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$.
- Other names may include the "$t$-test for matched pairs" or the "$t$-test for dependent samples."
- As $N_r$ increases, the sampling distribution of $W$ converges to a normal distribution.
-
- 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.