geometric series is a series in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric...

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 =...

arithmetico-geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green). The series summation...

}}x^{n}.} The Taylor series of any polynomial is the polynomial itself. The Maclaurin series of ⁠1/1 − x⁠ is the geometric series 1 + x + x 2 + x 3 + ⋯...

In mathematics, an infinite geometric series of the form ∑ n = 1 ∞ a r n − 1 = a + a r + a r 2 + a r 3 + ⋯ {\displaystyle \sum _{n=1}^{\infty...

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of positive real numbers by using the product...

can view power series as being like "polynomials of infinite degree", although power series are not polynomials. The geometric series formula 1 1 − x...

"values", one can justify that the series converges to 1/2. Treating Grandi's series as a divergent geometric series and using the same algebraic methods...

infinity. The value of the series is then b1 − L. An arithmetico-geometric series is a generalization of the geometric series, which has coefficients of...

physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry...

the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes...

generalization of a geometric series of real or complex numbers to a geometric series of operators. The generalized initial term of the series is the identity...

_{n=1}^{\infty }\left(1-(2i)^{n-1}\right)z^{-n}.} This series can be derived using geometric series as before, or by performing polynomial long division...

exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 20 + 21 + 22...

proved (for any integer t {\displaystyle t} ) by using the formula for geometric series: (using y = 1 − x {\displaystyle y=1-x} ) t = 1 + 1 + ⋯ + 1 ≥ 1 + y...

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series, S = I + A + A 2 + ⋯ + A n {\displaystyle S=I+A+A^{2}+\cdots...

In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points...

In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: The probability distribution...

integer values of α. The negative binomial series includes the case of the geometric series, the power series 1 1 − x = ∑ n = 0 ∞ x n {\displaystyle {\frac...

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative...

I. Motomura developed the geometric series model based on benthic community data in a lake. Within the geometric series each species' level of abundance...

geometric may also refer to: Geometric distribution of probability theory and statistics Geometric series, a mathematical series with a constant ratio between...

infinite series ⁠1/2⁠ + ⁠1/4⁠ + ⁠1/8⁠ + ⁠1/16⁠ + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1...

as a geometric series with the common ratio 1. For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2...

sum of the arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE...

geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra...

geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series,...

mathematics; it was used by Archimedes circa 250–200 BC. As it is a geometric series with first term ⁠1/4⁠ and common ratio ⁠1/4⁠, its sum is ∑ n = 1 ∞...

A Gabriel's horn (also called Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume. The name refers to...

The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio...