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:

    “... whilst you are proclaiming peace and good will to men, Emancipating all Nations, you insist upon retaining absolute power over wives. But you must remember that Arbitrary power is like most other things which are very hard, very liable to be broken—and notwithstanding all your wise Laws and Maxims we have it in our power not only to free ourselves but to subdue our Masters, and without violence throw both your natural and legal authority at our feet ...”
    —Abigail Adams (1744–1818)

    “Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
    —Jean Baudrillard (b. 1929)