A power series is a series of the form
with or . The are called the coefficients and the center of the power series. is called the constant term.
Where it converges the power series defines a function, which can thus be represented by a power series. This is what power series are usually used for. Every power series is convergent at least at where it converges to . In addition it is absolutely and uniformly convergent in the region , with
It is divergent for every with . For no general predictions can be made. If , the power series converges absolutely and uniformly for every real or complex The real number is called the radius of convergence of the power series.
Power series have some important :
If a power series converges for a then it also converges for all with .
Also, if a power series diverges for some then it diverges for all with .
For Power series can be added by adding coefficients and multiplied in the obvious way:
(Uniqueness) If two power series are equal and their are the same, then their coefficients must be equal.
Power series can be termwise differentiated and integrated. These operations keep the radius of convergence.
|Date of creation||2013-03-22 12:32:55|
|Last modified on||2013-03-22 12:32:55|
|Last modified by||azdbacks4234 (14155)|