8.6Power Series

Definition 8.6.1 If a0, a1, a2, . . . is a sequence of real numbers, then the

series P∞ akxk is called a power series in x. A positive number r is called

k=1

the radius of convergence and the interval (−r, r) is called the interval of convergence of the power series if the power series converges absolutely for all x in (−r, r) and diverges for all x such that |x| > r. The end point x = r is

included in the interval of convergence if P∞ akrk converges. The end point

k=1

x = −r is included in the interval of convergence if the series P∞ (−1)kakrk

k=1

converges. If the power series converges only for x = 0, then the radius of convergence is defined to be zero. If the power series converges absolutely for all real x, then the radius of convergence is defined to be ∞.

∞

X

Theorem 8.6.1 If the series cnxn converges for x = r 6= 0, then the

n=1

∞

X

series cnxn converges absolutely for all numbers x such that |x| < |r|.

n=0

∞

X

Proof. Suppose that cnrn converges. Then, by the Divergence Test,

n=0

lim cnrn = 0.

n→∞

For = 1, there exists some natural number m such that for all n ≥ m,

|cnrn| < = 1.

Let

M = max{|cnrn| + 1 : 1 ≤ n ≤ m}.

Then, for each x such that |x| < |r|, we get |x/r| < 1 and

n=0

x such that |x − a| < |r|. This completes the proof of the theorem.

∞

X

Theorem 8.6.3 Let cnxn be any power series. Then exactly one of the

n=0

following three cases is true.

(i)The series converges only for x = 0.

(ii)The series converges for all x.

(iii)There exists a number R such that the series converges for all x with |x| < R and diverges for all x with |x| > R.

Proof. Suppose that cases (i) and (ii) are false. Then there exist two

By Theorem 8.6.1, the series converges absolutely for all x such that |x| < |p|.

Let

∞

X

A = {p : cnpn converges}.

n=0

The set A is bounded from above by q. Hence A has a least upper bound, say

R. Clearly |p| ≤ R < q and hence R is a positive real number. Furthermore,

∞

X

cnxn converges for all x such that |x| < R and diverges for all x such that

n=0

|x| > R. We define R to be 0 for case (i) and R to be ∞ for case (ii). This completes the proof of Theorem 8.6.3.

∞

X

Theorem 8.6.4 Let cn(x − a)n be any power series. Then exactly one

n=0

of the following three cases is true:

(i)The series converges only for x = a and the radius of convergence is 0.

(ii)The series converges for all x and the radius of convergence is ∞.

(iii)There exists a number R such that the series converges for all x such that |x − a| < R and diverges for all x such that |x − a| > R.

converges absolutely for |x| < R.

Proof. For each x such that |x| < R, choose a number r such that |x| < r <

There exists some M such that |cnr | ≤ M for each natural number n. Then

n=0

X

such that |x−a| < R, then the series cn(x−a)n may be di erentiated with

n=0

respect to x any number of times and each of the di erential series converges for all x such that |x − a| < R.

∞

X

Proof. Let u = x−a. Then cnun converges for all u such that |u| < R. By

n=0

∞

X

Theorem 8.6.5, the series ncnun−1 converges for all u such that |u| < R.

n=1

This term-by-term di erentiation process may be repeated any number of times without changing the radius of convergence. This completes the proof of this theorem.

X

≤n cnann−1

n=1

derivatives of all orders for |x| < R that are obtained by successive term-by-

By the fundamental theorem of calculus, f0(x) = g(x). This completes the proof of this theorem.

Definition 8.6.2 The radius of convergence of the power series

∞

X

ak(x − a)k

k=1

is

(a)zero, if the series converges only for x = a;

(b)r, if the series converges absolutely for all x such that |x − a| < r and diverges for all x such that |x − a| > r.

(c)∞, if the series converges absolutely for all real number x.

If the radius of convergence of the power series in (x − a) is r, 0 < r < ∞, then the interval of convergence of the series is (a −r, a + r). The end points x = a + r or x = a − r are included in the interval of convergence if the