Convergence tests

Convergence tests are mathematical techniques used to determine whether a series or sequence converges (approaches a finite limit) or diverges (grows indefinitely or does not settle at any finite value). These tests are particularly important in the study of infinite series in calculus and analysis, as they help evaluate the behavior of sums of infinitely many terms.

Abel's test is a convergence test for series of the form \(\sum a_n b_n\), where \(a_n\) is a monotonic sequence that converges to 0, and \(\{b_n\}\) is a bounded sequence. The test is useful in determining the convergence of series when you are dealing with products of sequences.

The Alternating Series Test is a method used to determine the convergence of an alternating series. An alternating series is a series of the form: \[ \sum_{n=1}^{\infty} (-1)^{n} a_n \] where \( a_n \) is a sequence of positive terms. The test provides a way to conclude that the series converges under specific conditions.

Cauchy's convergence test, also known as the Cauchy criterion for convergence, is a method used to determine whether a sequence of real or complex numbers converges. This criterion is particularly useful because it provides a way to check for convergence without requiring knowledge of the limit to which the sequence converges.

Cauchy's limit theorem is a result in real analysis that provides conditions under which a sequence of real or complex numbers converges. Specifically, it deals with the concepts of convergence in the context of sequences.

The Cauchy condensation test is a convergence test used in the analysis of infinite series, particularly for series with non-negative terms. It provides a method to determine the convergence or divergence of a series based on the behavior of a related series.

The Dini test is a method used in mathematics, particularly in real analysis, to determine the convergence of a sequence of functions. More specifically, it is applicable to the study of pointwise convergence of a sequence of real-valued functions defined on a common domain. The test is based on the idea of comparing the behavior of the functions in the sequence with a "monotone" function or to establish some control over their convergence through the use of integrals.

The Direct Comparison Test is a method used in calculus to determine the convergence or divergence of an infinite series. It compares the series in question with a known benchmark series whose convergence behavior is already established. This test is particularly useful when dealing with series that have positive terms.

Dirichlet's test is a convergence test used primarily to determine the convergence of certain types of infinite series. It is particularly useful for series that may contain oscillatory components. The test is named after the German mathematician Johann Peter Gustav Lejeune Dirichlet.

The Integral Test for convergence is a method used to determine whether a series converges or diverges by comparing it to an improper integral. It applies specifically to series that consist of positive, decreasing functions. ### Statement of the Integral Test Let \( f(x) \) be a positive, continuous, and decreasing function for \( x \geq N \) (where \( N \) is some positive integer).

The Limit Comparison Test is a method used in calculus to determine the convergence or divergence of infinite series. It is particularly useful when the series in question is difficult to analyze directly. The test compares the given series with a known benchmark series.

The Nth-term test, also known as the Divergence Test, is a method used in calculus and series analysis to determine the convergence or divergence of an infinite series. It specifically applies to a series of the form: \[ \sum_{n=1}^{\infty} a_n \] where \(a_n\) is the nth-term of the series.

The Ratio Test is a method in mathematical analysis, particularly useful for determining the convergence or divergence of infinite series. It is often used for series whose terms involve factorials, exponentials, or other functions where the terms can grow rapidly. ### Statement of the Ratio Test Let \( \{a_n\} \) be a sequence of positive terms.

The Root Test is a method used to determine the convergence or divergence of an infinite series. Specifically, it helps assess the behavior of a series of the form: \[ \sum_{n=1}^{\infty} a_n \] where \( a_n \) is a sequence of real or complex numbers. The primary approach is based on the concept of the \( n \)-th root of the absolute value of the terms in the series.

The Stolz–Cesàro theorem is a result in real analysis that provides a method for evaluating limits of sequences, particularly those that appear in the context of quotients of two sequences. It is often used in situations where the conventional techniques for finding limits may not be easily applicable.