Power Series - Definition
A power series (in one variable 𝑥) is an infinite series of the form:
Indent
where:
- 𝑥 is the input variable
- 𝑎𝑛 represents the coefficient of the 𝑛th term
- 𝑐 is a constant
Power Series - SubTypes
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When centered on zero (i.e. 𝑐 = 0) it is a Maclaurin Series: | |
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When all coefficients (𝑎0, 𝑎1, …, 𝑎𝑛) equal the same value 𝑎, it is a Succession | |
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When expanding a function as an infinite series of terms expressed as the function’s derivatives at a single point, it is a Taylor Series: |
Power Series - Non-Examples
Power series does NOT permit:
- negative powers - for instance, 1 + 𝑥-1 + 𝑥-2 + … is not a power series (it is a Laurent series)
- fractional powers - for instance, 1 + 𝑥1/2 + 𝑥2/2 + 𝑥3/2+ … is not a power series (it is a Puiseux series)
- coefficients dependent on 𝑥 - for an instance, 𝑠𝑖𝑛(𝑥)𝑥 + 𝑠𝑖𝑛(2𝑥)𝑥2 + 𝑠𝑖𝑛(3𝑥)𝑥3 +… is not a power series
Power Series - Polynomial Example
Any polynomial can be easily expressed as a power series around any center 𝑐, although all but finitely many of the coefficients will be zero since a power series has infinitely many terms by definition.
For instance, the polynomial 𝑓(𝑥) = 𝑥2 + 2𝑥 + 3 can be written as a power series around the center:
- (𝑐 = 0) as: 𝑓(𝑥) = 3 + 2𝑥 + 1𝑥2 + 0𝑥3 + 0𝑥4 + …
- (𝑐 = 1) as: 𝑓(𝑥) = 6 + 4(𝑥 - 1) + 1(𝑥 - 1)2 + 0(𝑥 - 1)3 + 0(𝑥 - 1)4 + …
- 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
Power Series - Use Cases & Applications
- occur in combinatorics as generating functions (a kind of formal power series)
- occur in electronic 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 1⁄10
- occur in number theory, the concept of p-adic numbers is also closely related to that of a power series
Power Series - Subpages
- Power Series - Algebra (Addition／Subtraction／Multiplication／Division - Differentiation／Integration)
- Power Series - Radius of Convergence
- Power Series Expansion of Euler’s Equation／Formula
- Power Series Expansion of the Cosine Function
- Power Series Expansion of the Exponential Function
- Power Series Expansion of the Sine Function