Power Series

In mathematics, a power series (in one variable) is an infinite series of the form

where an represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). This series usually arises as the Taylor series of some known function; the Taylor series article contains many examples.

In many situations c is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots.

These power series arise primarily in analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at ⅟10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

Read more about Power Series:  Examples, Radius of Convergence, Analytic Functions, Formal Power Series, Power Series in Several Variables, Order of A Power Series

Famous quotes containing the words power and/or series:

    “Of a truth, Knowledge is power, but it is a power reined by scruple, having a conscience of what must be and what may be; whereas Ignorance is a blind giant who, let him but wax unbound, would make it a sport to seize the pillars that hold up the long- wrought fabric of human good, and turn all the places of joy as dark as a buried Babylon.”
    —George Eliot [Mary Ann (or Marian)

    “Every man sees in his relatives, and especially in his cousins, a series of grotesque caricatures of himself.”
    —H.L. (Henry Lewis)