"In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity."
A series is the sum of terms in an infinite sequence.
Sequences: A sequence is an ordered list of elements that are related to each other in some way. In calculus, we often use sequences to define and study series.
Convergence and Divergence: A series converges if the sum of its terms approaches a finite limit as the number of terms increases. A series diverges if the sum of its terms does not approach a finite limit.
Absolute Convergence: A series has absolute convergence if the sum of the absolute values of its terms converges.
Conditional Convergence: A series has conditional convergence if it is convergent, but not absolutely convergent.
Comparison Test: The comparison test is a way to determine whether a series converges or diverges by comparing it to a known convergent or divergent series.
Ratio Test: The ratio test is a way to determine whether a series converges or diverges by taking the ratio of consecutive terms.
Integral Test: The integral test is a way to determine whether a series converges or diverges by comparing it to the integral of a continuous and positive function.
Alternating Series Test: The alternating series test is a way to determine whether a series converges or diverges if its terms alternate in sign.
Power Series: A power series is a series in which the terms are powers of a variable x.
Taylor Series: A Taylor series is a type of power series that approximates a function as a sum of its derivatives.
Maclaurin Series: A Maclaurin series is a type of Taylor series in which the function is approximated around x=0.
Radius of Convergence: The radius of convergence is the distance between the center of a power series and the nearest point where it diverges.
Summation Notation: Summation notation is a shorthand way of writing a series using the sigma notation.
Geometric Series: A geometric series is a series in which each term is a constant multiple of the previous one.
Arithmetico-Geometric Series: An arithmetico-geometric series is a series that combines an arithmetic progression with a geometric progression.
Arithmetic Series: A series where each term is obtained by adding a constant value known as the common difference.
Geometric Series: A series where each term is obtained by multiplying the previous term by a constant value known as the common ratio.
Harmonic Series: A series where each term is the reciprocal of a natural number, i.e., 1/1, 1/2, 1/3, 1/4, ….
Alternating Series: A series where each term alternates between positive and negative values.
Telescoping Series: A series where most of the terms cancel out, leaving only a few terms.
Power Series: A series where each term is a multiple of the variable raised to different powers, like x, x^2, x^3, ….
Taylor Series: A power series that represents a function as an infinite sum of its derivatives, evaluated at a given point.
Maclaurin Series: A special case of the Taylor series, where the series is centered around x = 0.
Fourier Series: A series that represents a periodic function as a sum of sine and cosine functions.
Laurent Series: A series that represents a function as a sum of polynomial terms and terms with negative powers of the variable.
Dirichlet Series: A series that represents the distribution of prime numbers, expressed as a sum over the reciprocals of primes raised to different powers.
Hypergeometric Series: A series that solves a particular type of differential equation, where the ratio of the terms satisfies a certain recurrence relation.
"The study of series is a major part of calculus and its generalization, mathematical analysis."
"Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics, and finance."
"This paradox was resolved using the concept of a limit during the 17th century."
"Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums."
"Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist."
"Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series."
"Any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other."
"A series may be represented (or denoted) by an expression like ∑(a1, a2, a3, ...) or, using the summation sign, ∑ ai."
"If the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series."
"When the limit as n tends to infinity of the finite sums of the n first terms of the series exists, one says that the series is convergent or summable."
"Otherwise, the series is said to be divergent."
"The notation ∑i=1∞ ai denotes both the series—that is, the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process."
"Generally, the terms of a series come from a ring, often the field of the real numbers or the field of the complex numbers."
"In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term."
"The multiplication is the Cauchy product."
"Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions."
"Infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics, and finance."
"This paradox was resolved using the concept of a limit during the 17th century."
"The total time for Achilles to catch the tortoise is given by a series."