"In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity."
The study of infinite series and sequences, including convergence and divergence.
Arithmetic sequence: A sequence of numbers where each term is obtained by adding a constant value to the previous term.
Geometric sequence: A sequence of numbers where each term is obtained by multiplying the previous term by a constant value.
Series: An infinite sum of terms in a sequence.
Sequences and series in calculus: The study of sequences and series in the context of calculus.
Arithmetic sequence: A sequence of numbers where the difference between any two consecutive terms is constant.
Geometric sequence: A sequence of numbers where each term is the product of the previous term and a constant ratio.
Harmonic sequence: A sequence of numbers where each term is the reciprocal of a positive integer.
Fibonacci sequence: A sequence of numbers where each term is the sum of the two preceding terms.
Pascal's triangle: A triangle of numbers where each number is the sum of the two numbers above it.
Alternating series: A series where the signs of the terms alternate between positive and negative.
Convergent series: A series that has a finite sum.
Divergent series: A series that does not have a finite sum.
Infinite series: A series with an infinite number of terms.
Power series: A series of the form ∑(n=0)∞ a_n x^n, where x is a variable and the coefficients (a_n) are constants.
Taylor series: A power series that represents a function by expanding it around a point.
Fourier series: A series of the form ∑(n=-∞)∞ c_n e^(inx), where c_n are constants and e^(inx) is a complex exponential function.
Binomial series: A series that arises from expanding (1+x)^n using the binomial theorem.
Trigonometric series: A series of the form ∑(n=-∞)∞ a_n cos(nx) + b_n sin(nx), where a_n and b_n are constants.
Dirichlet series: A series of the form ∑(n=1)∞ a_n / n^s, where a_n are constants and s is a complex variable.
Zeta series: A Dirichlet series with a_1 = 1 and s = 2, which is used to define the Riemann zeta function.
Sequences of partial sums: A sequence of the partial sums of a series, which can be used to test whether a series converges or diverges.
Series expansions: Techniques for expressing functions as power series or other types of series.
Recurrence relations: Relationships between successive terms of a sequence, which can be used to define sequences or solve problems involving them.
Continued fractions: An expression of the form a_0 + 1/(a_1 + 1/(a_2 + 1/(a_3 + ...))), which can be used to represent numbers or solve certain types of equations.
"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."