Direct comparison testIn mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit 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.
Ratio testIn mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The usual form of the test makes use of the limit The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series diverges; 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.
AntiderivativeIn calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.
Convergence testsIn mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series . If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
Basel problemThe Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight.
Summation by partsIn mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Suppose and are two sequences. Then, Using the forward difference operator , it can be stated more succinctly as Summation by parts is an analogue to integration by parts: or to Abel's summation formula: An alternative statement is which is analogous to the integration by parts formula for semimartingales.
Integral test for convergenceIn mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Consider an integer N and a function f defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series converges to a real number if and only if the improper integral is finite. In particular, if the integral diverges, then the series diverges as well.
Alternating seriesIn mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not.
Cauchy productIn mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
Root testIn mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series. The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test.
