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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.
De analysi per aequationes numero terminorum infinitas (or On analysis by infinite series, [1] On Analysis by Equations with an infinite number of terms, [2] or On the Analysis by means of equations of an infinite number of terms) [3] is a mathematical work by Isaac Newton.
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]
A series or, redundantly, an infinite series, is an infinite sum. It is often represented as [ 8 ] [ 15 ] [ 16 ] a 0 + a 1 + a 2 + ⋯ or a 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,} where the terms a k {\displaystyle a_{k}} are the members of a sequence of numbers ...
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution).
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
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.
In the example above, for the maximal product measure, the diagonal has infinite measure so the double integral of |f| is infinite, and Fubini's theorem holds vacuously. However, if we give X × Y the product measure such that the measure of a set is the sum of the Lebesgue measures of its horizontal sections, then the double integral of | f ...
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