A series of the following form
is referred to as a power series about c or a power series centered at c. x
is the variable and an's are the coefficients of the series. The
sum of a power series can be represented using a function:
where the function f(x) resembles a polynomial.
A power series can either converge or diverge for a given value of x.
One can use the convergence testing methods introduced previously (e.g., ratio
test and root test) to determine if a power series is convergent
or divergent. The following theorem can also be applied to determine
the convergence of a power series:
Interval of Convergence
Geometric Series Σxn
For a given power series ,
only three possibilities exist:
The power series converges only when x = c.
(2) The power series converges for all x.
(3) There is a number R > 0 such that the series converges absolutely for ,
the series diverges for ,
where R is the radius of convergence. The interval of convergence for a power
series consists of all the values x of which the series is convergent.
For example, the geometric series Σxn is
a special case of the power series (an = 1 for all n and c
= 0). It was shown previously that
this series is convergent when -1 < x < 1 and it is divergent when .
Hence, the radius of convergence is R = 1 and the interval of convergence
is (-1, 1). The figure on the right shows the partial sum of the geometric
series Σ(1/2)n, and it converges
The following theorem is useful in differentiating and integrating a power
series (term-by-term differentiation and integration):
For a given power series with
a radius of convergence R > 0, the function
is continuous and differentiable on the interval (a - R, a + R) and
of Order 0 and 1
Power series is useful and important mainly because it can be used
to represent functions arise in mathematics, science and engineering
applications. One such function is called the Bessel function. The
Bessel function of order 0 is defined by
while the Bessel function of order 1 is defined as:
Note that J'0(x) = - J1(x). The Bessel function appears
when solving Kepler's equation for describing the motion of planets and when
dealing with heat conduction problems. Bessel functions Jo(x) and
J1(x) are plotted in the diagram on the left.