In mathematics, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the...
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A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by...
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Arithmetic progression (redirect from Arithmetic sequence)
\{1,4,7\}.} Geometric progression Harmonic progression Triangular number Arithmetico-geometric sequence Inequality of arithmetic and geometric means Primes...
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series – Sequence of equally spaced numbersPages displaying short descriptions of redirect targets Arithmetico-geometric sequence – Mathematical sequence satisfying...
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{da(1-a^{n-1})}{(1-a)^{2}}}\end{aligned}}} (sum of an arithmetico–geometric sequence) There exist very many summation identities involving binomial...
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Series (mathematics) (redirect from Summable sequence)
converges if the sequence bn converges to a limit L—as n goes to infinity. The value of the series is then b1 − L. An arithmetico-geometric series is a generalization...
70 KB (11,502 words) - 09:39, 21 October 2024
Curl (mathematics) (section Curl geometrically)
expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is...
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Continuity (mathematics) Absolute continuity Dini continuity Equicontinuity Geometric continuity Parametric continuity Classification of discontinuities Coarse...
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Taylor series (section Geometric series)
polynomial is the polynomial itself. The Maclaurin series of 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ . {\displaystyle 1+x+x^{2}+x^{3}+\cdots ...
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a n } {\displaystyle \{a_{n}\}} is a sequence of real numbers and { b n } {\displaystyle \{b_{n}\}} a sequence of complex numbers satisfying a n ≥ a...
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series ∑ n = 0 ∞ c n {\displaystyle \sum _{n=0}^{\infty }c^{n}} is the geometric series with common ratio c ∈ ( 0 ; 1 ) {\displaystyle c\in (0;1)} , hence...
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linking these ideas is that of homological integration (sometimes called geometric integration theory), pioneered by Georges de Rham and Hassler Whitney...
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decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even...
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smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms...
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monotonically, but this condition is not necessary for convergence. The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3. The alternating...
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rational number. The power rule for integrals was first demonstrated in a geometric form by Italian mathematician Bonaventura Cavalieri in the early 17th...
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if ( a n ) {\displaystyle (a_{n})} is a sequence of real numbers and ( b n ) {\displaystyle (b_{n})} a sequence of complex numbers satisfying ( a n ) {\displaystyle...
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topics. Limit (mathematics) Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous...
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then | a n | ≤ k n < 1 {\displaystyle |a_{n}|\leq k^{n}<1} . Since the geometric series ∑ n = N ∞ k n {\displaystyle \sum _{n=N}^{\infty }k^{n}} converges...
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citation needed] Craven, Thomas; Csordas, George (1977), "Multiplier sequences for fields", Illinois J. Math., 21 (4): 801–817, doi:10.1215/ijm/1256048929...
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Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts. The theorem can be derived as follows. For...
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}(x)} , then ‖ A − I ‖ < 1 / 2 {\displaystyle \|A-I\|<1/2} . Using the geometric series for B = I − A {\displaystyle B=I-A} , it follows that ‖ A − 1 ‖...
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series in his precalculus. Today's course may cover arithmetic and geometric sequences and series, but not the application by Saint-Vincent to gain his...
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=S_{n}+(T-T_{n})\leq T.} S n {\displaystyle S_{n}} is a nondecreasing sequence and S n + ( T − T n ) {\displaystyle S_{n}+(T-T_{n})} is nonincreasing...
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a_{n}} is a convergent series, b n {\displaystyle b_{n}} is a monotone sequence, and b n {\displaystyle b_{n}} is bounded. Then ∑ a n b n {\displaystyle...
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sequence of numbers ε i > 0 {\displaystyle \varepsilon _{i}>0} such that lim i ε i = 0 {\displaystyle \lim _{i}\varepsilon _{i}=0} , and a sequence x...
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a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers, the series ∑...
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the harmonic series raise the question of whether there are monotone sequences such that f(n) decreases to 0 faster than 1/n but slower than 1/n1+ε in...
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realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related. From...
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They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Infinitesimal calculus was...
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