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|power series |
A sum of successively higher integral powers of a variable or combination of variables, each multiplied by a constant coefficient.
in mathematics, an infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1+x+x2+x3+. Usually, a given power series will converge (that is, approach a finite sum) for all values of x within a certain interval around zero-in particular, whenever the absolute value of x is less than some positive number r, known as the radius of convergence. Outside of this interval the series diverges (is infinite), while the series may converge or diverge when x=r. The radius of convergence can often be determined by a version of the ratio test for power series: given a general power series a0+a1x+a2x2+,in which the coefficients are known, the radius of convergence is equal to the limit of the ratio of successive coefficients. Symbolically, the series will converge for all values of x such that