The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.Another geometric series (coefficient a = 4/9 and common ratio r = 1/9) shown as areas of purple squares. The total purple area is S = a / (1 - r) = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the unit square is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential.
The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms.
mathematics, a geometricseries is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series 1 2 + 1...
sequence's start value. The sum of a geometric progression's terms is called a geometricseries. The n-th term of a geometric sequence with initial value a =...
}}x^{n}.} The Taylor series of any polynomial is the polynomial itself. The Maclaurin series of 1/1 − x is the geometricseries 1 + x + x 2 + x 3 + ⋯...
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their...
can view power series as being like "polynomials of infinite degree," although power series are not polynomials. The geometricseries formula 1 1 − x...
the second part of a geometricseries. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes...
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...
_{n=1}^{\infty }\left(1-(2i)^{n-1}\right)z^{-n}.} This series can be derived using geometricseries 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 geometricseries is: 20 + 21 + 22...
Matrix polynomials can be used to sum a matrix geometricalseries as one would an ordinary geometricseries, S = I + A + A 2 + ⋯ + A n {\displaystyle S=I+A+A^{2}+\cdots...
{\displaystyle k} times repeated application. This generalizes the geometricseries. The series is named after the mathematician Carl Neumann, who used it in...
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 geometricseries, the power series 1 1 − x = ∑ n = 0 ∞ x n {\displaystyle {\frac...
sum of the arithmetic and geometricseries as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometricseries in his Kalpasūtra in 433 BCE...
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 mathematics, a geometric algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors...
I. Motomura developed the geometricseries model based on benthic community data in a lake. Within the geometricseries each species' level of abundance...
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