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:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)