"In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity."
An infinite sum of terms in a sequence.
Definition and Simple Examples: An introduction to what a series is and how it differs from a sequence.
Convergence and Divergence: The concept of a series converging, diverging, or oscillating, and how to determine which type a series is.
Absolute and Conditional Convergence: The differences between absolute and conditional convergence and how to identify each type.
Alternating Series: How to identify and understand alternating series and their convergence.
Ratio and Root Tests: Two common tests used to determine if a series converges or diverges.
Comparison Tests: How to use comparisons with known convergent or divergent series to determine the convergence of a new series.
Limit Comparison Tests: A variation of the comparison test that compares the ratio or limit of two series to determine convergence.
Tail Bounds and Error Estimates: How to estimate the error in approximating a series to a certain level of accuracy.
Power Series: An expansion of a function as an infinite sum of terms, typically used in calculus.
Taylor and Maclaurin Series: The use of power series to approximate functions as locally linear approximations.
Fourier Series: A representation of a periodic function as a sum of sines and cosines.
Infinite Products: The product analogue of a series, where terms are multiplied instead of added.
Applications of Series: Real-world applications of series, including their use in physics, engineering, and finance.
Series with Complex Terms: How to handle series with complex numbers and imaginary terms.
Dirichlet Series: A special type of infinite series used in number theory.
Zeta Function: A special type of Dirichlet series that is of particular interest in number theory and calculus.
Harmonic Series: A famous divergent series and its properties.
Alternating Harmonic Series: A famous convergent alternating series and its properties.
Periodic Sequences and Series: The study of sequences and series that exhibit periodic behavior.
Geometric Series: A type of series in which each term is a multiple of the previous term.
Arithmetic series: A sequence of numbers where each term is formed by adding a constant value to the previous term.
Geometric series: A sequence of numbers where each term is formed by multiplying the previous term by a common ratio.
Harmonic series: A series of numbers of the form 1/1 + 1/2 + 1/3 + ... + 1/n.
Alternating series: A series of numbers where the signs of consecutive terms alternate.
Telescoping series: A series whose partial sums cancel all but a finite number of terms.
Power series: A series of the form ∑anxn, where a0, a1, a2, …, an are constants and x is a variable.
Taylor series: A power series representing a function as a sum of infinitely many terms that involve the function's derivatives evaluated at a point.
Fourier series: A representation of a periodic function as a sum of sinusoidal functions.
Dirichlet series: A series of the form ∑an/n^s, where s is a complex variable and a0, a1, a2, …, an are constants.
Laurent series: A power series involving negative powers of the variable, often used to represent complex functions.
"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."