In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions.
Read more about Formal Power Series: Introduction, The Ring of Formal Power Series, Applications, Interpreting Formal Power Series As Functions, Examples and Related Topics
Famous quotes containing the words formal, power and/or series:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“A photo of someone else’s childhood,
a garden in another country—world
he had no part in and has no power to imagine:
yet the old man who has failed his memory
keens over the picture— ‘Them happy days—
gone—gone for ever!’”
—Denise Levertov (b. 1923)
“Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.”
—Sonia Taitz (20th century)