“power series”的意思、由来-百科全书

power series

mathematics

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 (convergence) (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

For instance, the infinite series 1+x+x²+x³+⋯ has a radius of convergence of 1 (all the coefficients are 1)—that is, it converges for all −1\\<x\\<1—and within that interval the infinite series is equal to 1/(1−x). Applying the ratio test to the series

1+x/1!+x²/2!+x³/3!+⋯

(in which the factorial notation n! means the product of the counting numbers from 1 to n) gives a radius of convergence of

so that the series converges for any value of x.

Most functions can be represented by a power series in some interval (see table-->

). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (x−c), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations (differential equation).

词条	power series
释义	power series mathematics 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 (convergence) (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 For instance, the infinite series 1+x+x²+x³+⋯ has a radius of convergence of 1 (all the coefficients are 1)—that is, it converges for all −1\\<x\\<1—and within that interval the infinite series is equal to 1/(1−x). Applying the ratio test to the series 1+x/1!+x²/2!+x³/3!+⋯ (in which the factorial notation n! means the product of the counting numbers from 1 to n) gives a radius of convergence of so that the series converges for any value of x. Most functions can be represented by a power series in some interval (see table-->). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (x−c), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations (differential equation).
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