"In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity."
A sequence of the partial sums of a series, which can be used to test whether a series converges or diverges.
Sequences: A sequence is an ordered list of elements or terms. In the context of partial sums, a sequence represents the sum of a series up to a certain point.
Series: A series is an infinite sum of terms, where each term is added to the previous one.
Convergence and Divergence: Convergence and divergence are properties that describe whether a series or sequence approaches a limit or not.
Geometric series: A geometric series is a series where each term is a constant multiple of the preceding term.
Telescoping series: A telescoping series is a series where most terms cancel out, leaving only a finite number of terms.
Absolute convergence: A series is said to be absolutely convergent if the series obtained by taking the absolute values of its terms converges.
Ratio test: The ratio test is a test used to determine the convergence or divergence of a series.
Integral test: The integral test is a test used to determine the convergence or divergence of a series by comparing it to an integral.
Limit comparison test: The limit comparison test is a test used to compare the convergence or divergence of two series.
Alternating series: An alternating series is a series where the signs of the terms alternate.
Power series: A power series is a series of the form ∑ a_n (x-a)^n.
Taylor series: A Taylor series is a power series representing a function as a sum of its derivatives.
Maclaurin series: A Maclaurin series is a Taylor series centered at 0.
Radius and interval of convergence: The radius and interval of convergence determine the range of values of x for which a power series converges.
Taylor's inequality: Taylor's inequality is used to approximate a function using its Taylor series.
Arithmetic sequence of partial sums: A sequence in which each term is the sum of previous term and a constant d.
Geometric sequence of partial sums: A sequence in which each term is the product of the previous term and a constant r.
Arithmetic-geometric sequence of partial sums: A sequence in which each term is the sum or product of an arithmetic and geometric sequence.
Fibonacci sequence: A sequence in which each term is the sum of the two preceding terms.
Harmonic sequence: A sequence in which each term is the reciprocal of the corresponding positive integer.
Power series: A series where each term is a constant multiple of an integer power of a variable.
Alternating series: A series where each term is positive or negative alternately.
Convergent series: A series that is finite and has a sum.
Divergent series: A series that has no finite sum.
Conditional convergent series: A series that converges, but its rearrangement converges to a different value.
Absolutely convergent series: A series that converges and its rearrangement converges to the same value.
Telescoping series: A series in which the terms cancel out in pairs, leaving only a finite number of terms.
Binomial series: A power series with binomial coefficients as its coefficients.
Exponential series: A power series with exponential functions as its coefficients.
Hypergeometric series: A power series with a ratio of two consecutive terms being a rational function of n.
"The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of 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 divided the race into infinitely many sub-races... which gives the time necessary for Achilles to catch up with the tortoise."
"...any (ordered) infinite sequence (a1, a2, a3, ...) of terms defines a series, which is the operation of adding the ai one after the other."
"To emphasize that there are an infinite number of terms, a series may be called an infinite series."
"...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."
"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 R of the real numbers or the field C of the complex numbers."
"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."
"When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent."
"...the expression ∑i=1∞ai is a generalization of the similar convention of denoting by a+b both the addition—the process of adding—and its result—the sum of a and b."
"Generally, the terms of a series come from a ring, often the field R of the real numbers..."
"...often the field C 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, and the multiplication is the Cauchy product."