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However, Abel's theorem states that if the series is convergent for some value z such that | z – c | = r, then the sum of the series for x = z is the limit of the sum of the series for x = c + t (z – c) where t is a real variable less than 1 that tends to 1.
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.
Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is =, has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function f(z) of Example 1 is the derivative of g(z). Example 3: The power series
The particular form of the Jacobi-type continued fractions (J-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to z for some specific, application-dependent component sequences, {ab i} and {c i}, where z ≠ 0 denotes the formal variable in the second power series ...
In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series. As an example, consider the initial value problem ″ + ′ + ′ =; = , ′ = which describes a solution to capillary-driven flow in ...
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for +. At z = 1 {\\displaystyle z=1} the series is equal to 1 − 1 + 1 − 1 + ⋯ , {\\displaystyle 1-1+1-1+\\cdots ,} but 1 1 + 1 = 1 2 . {\\displaystyle {\\tfrac ...