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Sum of an (infinite) geometric progression
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
Geometric_series
Mathematical sequence of numbers
ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series. Because each two successive numbers in the progression
Geometric_progression
Mathematical sequence satisfying a specific pattern
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
Arithmetico-geometric sequence
Arithmetico-geometric_sequence
Mathematical approximation of a function
allow the Taylor series of functions, such as the arctangent, to be computed in terms of simpler series, such as the geometric series. Several methods
Taylor_series
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
Divergent_geometric_series
Infinite sum
{\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
Series_(mathematics)
Inequality about exponentiations of ''1+x''
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
Bernoulli's_inequality
Topics referred to by the same term
geometric may also refer to: Geometric distribution of probability theory and statistics Geometric series, a mathematical series with a constant ratio between
Geometry_(disambiguation)
Method of differentiating single-term polynomials
{p}{q}}-a^{\frac {p}{q}}}{b-a}}\\[4pt]\end{aligned}}} Now, consider the geometric sum formula, b n − a n b − a = ∑ i = 0 n − 1 b ( n − 1 ) − i a i {\displaystyle
Power_rule
Infinite sum of monomials
power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense. The geometric series formula
Power_series
Infinite series summing alternating 1 and -1 terms
"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
Grandi's_series
Geometric treatise by Archimedes
the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes
Quadrature_of_the_Parabola
Concept in ecology
I. Motomura developed the geometric series model based on benthic community data in a lake. Within the geometric series each species' level of abundance
Relative_species_abundance
Mathematical series
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
Neumann_series
Mathematical problem
as to introduce exponents, zero power, capital-sigma notation, and geometric series. Updated for modern times using pennies and a hypothetical question
Wheat_and_chessboard_problem
Branch of mathematics
understood as geometric objects since Klein's Erlangen programme. Geometric group theory studies group actions on objects that are regarded as geometric (significantly
Geometry
Polynomial with a matrix as variable
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
Matrix_polynomial
Power series with negative powers
_{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
Laurent_series
Infinite series summable to 1
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
1/2_+_1/4_+_1/8_+_1/16_+_⋯
Divergent sum of positive unit fractions
different series, marked the first appearance of infinite series other than the geometric series in mathematics. However, this achievement fell into obscurity
Harmonic_series_(mathematics)
In mathematics, series built from equally spaced terms of another series
character) . In general, the bisections of a series are the even and odd parts of the series. Consider the geometric series ∑ n = 0 ∞ z n = 1 1 − z for | z |
Series_multisection
Point minimizing sum of distances to given points
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
Geometric_median
Infinite series that diverges
mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term
1_+_2_+_4_+_8_+_⋯
Better to receive money now than later
geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series,
Time_value_of_money
System for defining and representing engineering tolerances
Geometric dimensioning and tolerancing (GD&T) is a system for defining and communicating engineering tolerances via a symbolic language on engineering
Geometric dimensioning and tolerancing
Geometric_dimensioning_and_tolerancing
Geometric figure which has infinite surface area but finite volume
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
Gabriel's_horn
Divergent series
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
1_+_1_+_1_+_1_+_⋯
N-th root of the product of n numbers
In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection
Geometric_mean
Algebraic expansion of powers of a binomial
(1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .} which is the geometric series sum formula for the convergent case |x| < 1, whose common ratio is
Binomial_theorem
Branch of mathematics
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
Mathematical_analysis
Negative integer two units from the origin in mathematics
complex numbers. The sum of the powers of negative two is a divergent geometric series. Although divergent, its generalized sum is 1 3 {\displaystyle {\frac
−2
Approximation for factorials
{1}{5(2n+1)^{4}}}+\dots } To calculate the sum, we force them to be a geometric series: l n − l n + 1 < 1 3 ( 2 n + 1 ) 2 ∑ k = 0 ∞ [ 1 ( 2 n + 1 ) 2 ] k
Stirling's_approximation
Regularity in sensory qualia or abstract ideas
well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper
Pattern
Infinite series summable to 1/3
infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. It is a geometric series whose first
1/2_−_1/4_+_1/8_−_1/16_+_⋯
Mathematical series
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
Binomial_series
Characterization of even perfect numbers
that if a finite geometric series beginning at 1 with ratio 2 has a prime sum q, then this sum multiplied by the last term t in the series is perfect. Expressed
Euclid–Euler_theorem
Alternative decimal expansion of 1
000 = 0, and so ...999 = −1. Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers
0.999...
Summation method for divergent series
number (unless the bound on the error is made smaller). Consider the geometric series A ( z ) = ∑ k = 0 ∞ z k , {\displaystyle A(z)=\sum _{k=0}^{\infty }z^{k}
Borel_summation
Hindu astronomy, mathematics, science school in India
provided what is now considered the first example of a power series (apart from geometric series). Islamic scholars nearly developed a general formula for
Kerala school of astronomy and mathematics
Kerala_school_of_astronomy_and_mathematics
Arithmetic mean is greater than or equal to geometric mean
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
AM–GM_inequality
Operation on formal power series
{C} } defined such that | c z | < 1 {\displaystyle |cz|<1} , let the geometric series over the non-negative integral powers of ( c z ) n {\displaystyle (cz)^{n}}
Generating function transformation
Generating_function_transformation
Infinite series summable to 1/3
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 ∞
1/4_+_1/16_+_1/64_+_1/256_+_⋯
Function defined by a hypergeometric series
{3}}}}\end{aligned}}} When a = 1 and b = c, the series reduces into a plain geometric series, i.e. 2 F 1 ( 1 , b ; b ; z ) = 1 F 0 ( 1 ; ; z ) = 1
Hypergeometric_function
Mathematical power series of arctangent
Maclaurin series for x ↦ arctan ′ x = 1 / ( 1 + x 2 ) {\textstyle x\mapsto \arctan 'x=1{\big /}\left(1+x^{2}\right)} is a geometric series: 1 1 + x 2
Arctangent_series
Measure of algorithmic complexity
number of descriptions of length not exceeding n − c is given by the geometric series: 1 + 2 + 22 + ... + 2n − c = 2n−c+1 − 1. There remain at least 2n −
Kolmogorov_complexity
Graphical aid for deriving some concepts in combinatorics
1+1x+1x^{2}+1x^{3}+\ldots =1+x+x^{2}+x^{3}+\ldots ={\frac {1}{1-x}}.} The series is a geometric series, and the last equality holds analytically for |x| < 1, but is
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
expansion of the continued fraction matches Eq. 1 term-by-term. The geometric series formula 1 1 − u = 1 + u + u 2 + u 3 + … {\displaystyle {\frac
Proof_that_pi_is_irrational
List of mathematical contexts in which exponentiated terms are summed
integers can be expressed in this form. The sum of the terms in the geometric series is ∑ i = k n z i = z k − z n + 1 1 − z . {\displaystyle \sum _{i=k}^{n}z^{i}={\frac
Sums_of_powers
Greek mathematician and physicist (c. 287 – 212 BC)
figure at right, expressing the solution to the problem as an infinite geometric series with the common ratio 1/4: ∑ n = 0 ∞ 4 − n = 1 + 4 − 1 + 4 − 2 +
Archimedes
Traditional English riddle
interpretation that there are seven knives per loaf.) The problem is to sum the geometric series 7 + 7 2 + 7 3 + 7 4 + 7 5 + 7 6 = 137 256. {\displaystyle
As_I_was_going_to_St_Ives
Mathematical series with a finite sum
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
Convergent_series
Infinite series that is not convergent
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 ∞
Divergent_series
Product of numbers from 1 to n
the total time for these steps at all levels of recursion adds in a geometric series to O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} . The time for
Factorial
Number of partitions of an integer
expanding each factor 1 / ( 1 − x k ) {\displaystyle 1/(1-x^{k})} into the geometric series ( 1 + x k + x 2 k + x 3 k + ⋯ ) . {\displaystyle (1+x^{k}+x^{2k}+x^{3k}+\cdots
Partition function (number theory)
Partition_function_(number_theory)
{\displaystyle z=1} . The example of the geometric series gives an isolated singularity. An example of a series with a non-isolated singularity at z = 1
Vivanti–Pringsheim_theorem
Infinite series that diverges
4 − 8 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its
1_−_2_+_4_−_8_+_⋯
{\displaystyle \sum _{k=m}^{n}z^{k}={\frac {z^{m}-z^{n+1}}{1-z}}} (geometric series) ∑ k = 0 n z k = 1 − z n + 1 1 − z {\displaystyle \sum _{k=0}^{n}z^{k}={\frac
List_of_mathematical_series
Optimization method
observing that the times for the recursive subproblems decrease in a geometric series. In particular, Megiddo himself used this approach in his linear time
Prune_and_search
Infinite products of functions indexed by primes
that in which a(n) is totally multiplicative, so that P(p, s) is a geometric series. Then P ( p , s ) = 1 1 − a ( p ) p s , {\displaystyle P(p,s)={\frac
Euler_product
Speed of convergence of a mathematical sequence
any geometric series to its limit has error terms that are equal to a geometric progression, so similar relationships hold among geometric series as well
Rate_of_convergence
Marketing concept
monthly profit per customer, and dividing by the churn rate sums the geometric series representing the chance the customer will still be around in future
Customer_lifetime_value
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra
Geometric_algebra
Compounding sum paid for the use of money
{\displaystyle M'=\sum _{i=0}^{t-1}{M(1+r)^{t-i}}} Recognizing the geometric series: M ′ = M ∑ i = 0 t − 1 ( 1 + r ) t 1 ( 1 + r ) i {\displaystyle M'=M\sum
Compound_interest
1998 mathematics textbook
1998 as volume 81 of their Undergraduate Texts in Mathematics book series. Geometric Constructions has ten chapters. The first two discuss straightedge
Geometric_Constructions
Description of the behaviour of bosons
Thus, the grand partition function for bosons can be considered a geometric series and may be evaluated as such: Z = ∑ N = 0 ∞ exp ( N ( μ − ε ) / k
Bose–Einstein_statistics
Taylor series for the natural logarithm
{\displaystyle x+1} . Alternatively, one can start with the finite geometric series ( t ≠ − 1 {\displaystyle t\neq -1} ) 1 − t + t 2 − ⋯ + ( − t ) n −
Mercator_series
Expression in commutative algebra
formal geometric series that is a factor in the middle expression. The identity can be justified by considering how the product of those geometric series is
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
Decimal representation of a number whose digits are periodic
{1}{1000}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}} The above series is a geometric series with the first term as 1/10 and the common factor 1/10. Because
Repeating_decimal
Analytic function in mathematics
\,} 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
Lacunary_function
by bringing them in a row. (For example, this will be needed in a geometric series with ratio − 4 {\displaystyle -4} .) This process of successive averaging
Van Wijngaarden transformation
Van_Wijngaarden_transformation
Probability distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: The probability distribution
Geometric_distribution
Roughly, the number of k-dimensional holes on a topological surface
polynomial but rather an infinite series 1 + x 2 + x 4 + ⋯ {\displaystyle 1+x^{2}+x^{4}+\dotsb } , which, being a geometric series, can be expressed as the rational
Betti_number
Class of functions in transcendental number theory
of f grow no faster than a geometric series. Indeed, the functions can be considered as generalisations of geometric series, whence the name G-function
Siegel_G-function
Development of mathematics in South Asia
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
Indian_mathematics
Mathematical criterion about whether a series converges
_{n=1}^{\infty }2^{n-n\alpha }=\sum _{n=1}^{\infty }2^{(1-\alpha )n}} (ii) is a geometric series with ratio 2 ( 1 − α ) {\displaystyle 2^{(1-\alpha )}} . (ii) is finitely
Convergence_tests
Geometric pattern characteristic of Muslim art
Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create
Islamic_geometric_patterns
English saying meaning "equivalent retaliation"
. . , {\displaystyle 6+6\delta +6\delta ^{2}+6\delta ^{3}...,} a geometric series summing to 6 1 − δ {\displaystyle {\frac {6}{1-\delta }}} If a player
Tit_for_tat
Divergent series
literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4
1_+_2_+_3_+_4_+_⋯
Extension of the domain of an analytic function (mathematics)
z {\displaystyle f(z)=1/z} (because it is a geometric series), and focus on recentering the power series at a different point a ∈ U {\displaystyle a\in
Analytic_continuation
Casino game of chance
betting systems are essentially an attempt to beat the fact that a geometric series with initial value of 0.95 (American roulette) or 0.97 (European roulette)
Roulette
Linear transform from the time domain to the frequency domain
{1}{1-0.5\,z^{-1}}}.} The last equality arises from the infinite geometric series and the equality only holds if | 0.5 z − 1 | < 1 {\displaystyle \vert
Z-transform
Branch of discrete mathematics
which dates to the 16th century BC. The problem concerns a certain geometric series, and has similarities to Fibonacci's problem of counting the number
Combinatorics
Idea that small causes can have large effects
finite predictability was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing
Butterfly_effect
Every polynomial has a real or complex root
{\displaystyle \int _{c(r)}R(z)\,dz=0.} On the other hand, R(z) expanded as a geometric series gives: R ( z ) = z − 1 ( I n − z − 1 A ) − 1 = z − 1 ∑ k = 0 ∞ 1 z
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
French mathematician and lawyer (1601–1665)
With his method, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when
Pierre_de_Fermat
Point where a person ceases employment permanently
the standard mathematical formula for the sum of a geometric series. (Or if ireal =0 then the series in braces sums to p since it then has p equal terms)
Retirement
Representation theory
{\sin((m+1)\theta )}{\sin \theta }}.} The character in this case is a geometric series with R = e 2 i θ {\displaystyle R=e^{2i\theta }} and that preceding
Weyl_character_formula
American painter
terrain she is exploring today." The Self/Portraits series pairs Gold's painterly sense for geometric abstraction with the entrance of representational
Sharon_Gold
Markov chain in which all states can be absorbing
matrix. The computation of this formula is the matrix equivalent of the geometric series of scalars, ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle {\textstyle \sum
Absorbing_Markov_chain
Provides integral formulas for all derivatives of a holomorphic function
can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series applied to f ( ζ ) = 1 2 π i ∫
Cauchy's_integral_formula
Base of natural logarithms
which our series [...] is larger [than]. […] if a = b, [the lender] will be owed more than 21/2a and less than 3a. If a = b, the geometric series reduces
E_(mathematical_constant)
Belgian Jesuit and mathematician (1584–1667)
the summation of geometric series." He also resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus
Grégoire_de_Saint-Vincent
Algorithm for the kth smallest element in an array
set decreases in size exponentially and by induction (or summing the geometric series) one sees that performance is linear, as each step is linear and the
Quickselect
Infinite sum that is considered independently from any notion of convergence
\ \ n\geq 1.\end{aligned}}} An important special case is that the geometric series formula is valid in R [ [ X ] ] {\displaystyle R[[X]]} : ( 1 − X )
Formal_power_series
Series whose partial sums eventually only have a fixed number of terms after cancellation
} Every series is a telescoping series of its own partial sums. The product of a finite geometric series with initial term a {\displaystyle a} and
Telescoping_series
Formal power series
the geometric series ∑ n = 0 ∞ x n = 1 1 − x . {\displaystyle \sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.} The left-hand side is the Maclaurin series expansion
Generating_function
Infinite series whose terms alternate in sign
The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does
Alternating_series
Approximation of a function by a polynomial
This result is based on comparison with a geometric series, and the same method shows that if the power series based on a converges for some b ∈ R, it must
Taylor's_theorem
Number, approximately 1.3247
In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x + 1. Its decimal expansion begins
Plastic_ratio
Mathematical function
t/(t^{2}+z^{2})} as a geometric series and substituting an integral representation of the Bernoulli numbers leads to the same asymptotic series as above. Furthermore
Digamma_function
travel, tourism, insurance
GEOMETRIC SERIES
GEOMETRIC SERIES
Male
German
Old German name, GOMERIC means "man-power."
Girl/Female
Bengali, Hindu, Indian
Series of Leaves; Beauty of a Leaf
Girl/Female
Tamil
Shrankhla | à®·à¯à®°à®‚கலா
Born in the month of Shravan, Series
Shrankhla | à®·à¯à®°à®‚கலா
Male
Welsh
Welsh Arthurian legend name of the giant father of the beautiful Olwen. He was cursed to die if his daughter ever married. He lived in a magic castle that seemed to get farther away the closer one came to it. When Culhwch came to seek Olwen's hand, Ysbaddaden required that he complete a series of nearly impossible tasks before he would grant permission for them to marry. Meaning unknown.
Girl/Female
Tamil
Shrinkhla | à®·à¯à®°à¯€à®¨à¯à®•à®²à®¾
Series
Shrinkhla | à®·à¯à®°à¯€à®¨à¯à®•à®²à®¾
Girl/Female
Hindu
Series
Girl/Female
Tamil
Chitramala | சிதà¯à®°à®®à®¾à®²à®¾
Series of pictures
Chitramala | சிதà¯à®°à®®à®¾à®²à®¾
Girl/Female
Bengali, Indian
A Series of Leaves
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Sindhi, Telugu
Series of Pictures
Girl/Female
Tamil
Shrinkhala | à®·à¯à®°à¯€à®¨à¯à®•à®¾à®²à®¾
Born in the month of Shravan, Series
Shrinkhala | à®·à¯à®°à¯€à®¨à¯à®•à®¾à®²à®¾
Girl/Female
Hindu
Born in the month of Shravan, Series
Girl/Female
Hindu
Born in the month of Shravan, Series
Boy/Male
Greek
Greek surname. Euclid was an early developer of geometry theories.
GEOMETRIC SERIES
GEOMETRIC SERIES
GEOMETRIC SERIES
GEOMETRIC SERIES
GEOMETRIC SERIES
GEOMETRIC SERIES
GEOMETRIC SERIES
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