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the sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} is summable, and otherwise, when the limit does not exist, the series is...

a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have...

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the lim {\displaystyle \lim }...

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle...

defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series. This special case of a matrix...

mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential...

series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series 1...

called the sequence of partial sums of the series ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} . If the sequence of partial sums converges, then...

probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges...

converging sequence the sequence of the arithmetic means of its first n {\displaystyle n} members converges against the same limit as the original sequence, that...

introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials...

Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci...

the initial value. The sum of a geometric progression's terms is called a geometric series. The nth term of a geometric sequence with initial value a =...

of an infinite series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} . If the limit of the summand is undefined or nonzero, that is lim n →...

arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various...

limit of Riemann sums taken by sampling the function f in a set of points chosen from a fine partition of the interval. Therefore, if some sequence is...

mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045...

depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often...

difference of two consecutive terms of a sequence ( a n ) {\displaystyle (a_{n})} . As a consequence the partial sums of the series only consists of two terms...

Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating...