Examples
Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero. For instance, the polynomial can be written as a power series around the center as
or around the center as
or indeed around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.
The geometric series formula
which is valid for, is one of the most important examples of a power series, as are the exponential function formula
and the sine formula
valid for all real x. These power series are also examples of Taylor series.
Negative powers are not permitted in a power series, for instance is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as are not permitted (but see Puiseux series). The coefficients are not allowed to depend on, thus for instance:
- is not a power series.
Read more about this topic: Power Series
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)