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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, [5] and gave a remarkably accurate approximation of π. [80] [81]
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: = = + + + + +. The first n {\displaystyle n} terms of the series sum to approximately ln n + γ {\displaystyle \ln n+\gamma } , where ln {\displaystyle \ln } is the natural logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0.577 ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
This category is for mathematical identities, i.e. identically true relations holding in some area of algebra (including abstract algebra, or formal power series). Subcategories This category has only the following subcategory.
Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. [6] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below) .
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
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