In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant...

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

}}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 + ⋯...

{\displaystyle n} th truncation error of the infinite series. An example of a convergent series is the geometric series 1 + 1 2 + 1 4 + 1 8 + ⋯ + 1 2 k + ⋯ . {\displaystyle...

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...

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...

understood as geometric objects since Klein's Erlangen programme. Geometric group theory studies group actions on objects that are regarded as geometric (significantly...

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

power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense. The geometric series formula...

_{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...

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...

as to introduce exponents, zero power, capital-sigma notation, and geometric series. Updated for modern times using pennies and a hypothetical question...

different series, marked the first appearance of infinite series other than the geometric series in mathematics. However, this achievement fell into obscurity...

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

our series [a geometric series] is larger [than]. ... if a=b, [the lender] will be owed more than ⁠2+1/2⁠a and less than 3a.) If a=b, the geometric series...

"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...

\,} The power series converges locally uniform on any open domain |z| < 1. This can be proved by comparing f with the geometric series, which is absolutely...

our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) If a = b, the geometric series reduces...

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

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...

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...

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

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...

summation methods give the same answer for certain series. For instance, whenever r ≠ 1, the geometric series G ( r , c ) = ∑ k = 0 ∞ c r k = c + ∑ k = 0 ∞...

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...

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...

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...

provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of...

i ) − n 1 − ( 1 + i ) − 1 ) by using the equation for the sum of a geometric series = 1 − ( 1 + i ) − n 1 + i − 1 = 1 − ( 1 1 + i ) n i , {\displaystyle...