Power Series

Definition of a Power Series

We now investigate a generalization of polynomials.  The polynomial below is a fifth degree polynomial

3x5 - 2x4 + 5x3 + 4x - 1

if instead of a fifth degree polynomial, we consider a polynomial of infinite degree.

 Definition of a Power Series Let f(x) be the function represented by the series         Then f(x) is called a power series function.

More generally, if f(x) is represented by the series

Then we call f(x) a power series centered at x = c.

Examples

The power series

is centered at 0.

The power series

is centered at -2.

Next we want to investigate the domain of power series.  Recall to find the domain, we ask what values of x can the function handle?  This is particularly important with power series, since infinite series often do not converge.  It would be an insurmountable task to plug in each value of x and see if the series converges for that value.  Fortunately, we have the following theorem.

Theorem

If

is a power series centered at c then only the following three are possibilities for the domain of f.

1. The domain is the value c only.

2. There domain is all real numbers.

3. There exists a real number R such that all values of x that satisfy

| x - c |  <  R

are in the domain and values that satisfy

| x - c |  >  R

are not in the domain.

R is called the radius of convergence of f.

To compute the radius of convergence, we use the ratio test.

Example:
Find the radius of convergence of

Solution

We use the Ratio Test:

We solve

or
|x - 3| < 2

so that

1 < x < 5

Since

1/2(5 - 1) = 2

the radius of convergence is 2

Exercise:

Find the radius of convergence of

Taylor and Maclaurin Series

Since power series are functions, a natural question to ask is, "Can our everyday functions be represented as power series?"  Also, "Given a power series, can we find an everyday function that is equivalent to the power series?"

The following definition helps to answer these questions.

 Definition    The Taylor Series for f(x) centered at x = c is If c = 0, then the series is called the Maclaurin series for f.

We use the notation f (n) to denote the nth derivative of f.

Example:

Find the McLaurin Series  for

f(x) = cos(x)

Solution

We compute:

f(0) = 1           f '(0) = 0        f ''(0) = -1        f (3)(0) = 0

f (4)(0) = 1      f (5)(0) = 0       f (6)(0) = -1        f (7)(0) = 0

f (8)(0) = 1      f (9)(0) = 0        f (10)(0) = -1        f (11)(0) = 0

Hence we have the series

1 - x2 /2+ x4/4! - x6/6!  + x8/8! - x8/8!  + ...

We see that the series is

Exercises  Find the Taylor series expansion for

1. sin(x) centered at x = 0

2. ln(x) centered at x = 0

Differentiation  and Integration of Power Series

Since a power series is a function, it is natural to ask if the function is continuous, differentiable or integrable.  The following theorem answers this question.

 Theorem: Suppose that a function is given by the power series            f(x) =San(x - c)n and that the interval of convergence is            (c - R,c + R) (plus possible endpoints)   then f(x) is continuous, differentiable, and integrable on that interval (not necessarily including the endpoints).  To obtain the derivative or the integral of f(x) we can pass the derivative or integral through the S.  In other words                and             Furthermore, the radius of convergence for the derivative and integral is R.

Example:

Consider the series

f(x) = Sxn

by the GST this series converges for |x| < 1,  hence the center of convergence is 0 and the radius is 1.  By the  above theorem

f '(x) = Snxn-1

has center of convergence 0 and radius of convergence 1 also.  We can also say that

also has center of convergence 0 and radius of convergence 1.

Exercise:

Show that

satisfies the differential equation

y'' + xy' - y = 0

We can also use substitution to find power series.

Example

Find the Maclaurin series for

1

1 - x2

Solution

Substituting x2 for x in

we have

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