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1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. In mathematics, the 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. In summation notation ...
t. e. In mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
The simplest example is attempting to integrate the 1-form dx over the interval [0, 1]. Assuming the usual distance (and thus measure) on the real line, this integral is either 1 or −1 , depending on orientation: ∫ 0 1 d x = 1 {\displaystyle \textstyle {\int _{0}^{1}dx=1}} , while ∫ 1 0 d x = − ∫ 0 1 d x = − 1 {\displaystyle ...
Partial fraction decomposition. In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions ...
v. t. e. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations ...
The same illustration for The midpoint method converges faster than the Euler method, as . Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to ...
The Maurer–Cartan form[1] gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on P satisfying an integrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain ...
Constant of integration. In calculus, the constant of integration, often denoted by (or ), is a constant term added to an antiderivative of a function to indicate that the indefinite integral of (i.e., the set of all antiderivatives of ), on a connected domain, is only defined up to an additive constant. [1][2][3] This constant expresses an ...