Section 8.3, Slide 3Up to this point we have dealt with infinite series whose terms were fixed numbers.. The simplest kind is known as a power series, and the main question will be deter
Trang 1Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 8.3, Slide 1
Chapter 8
Infinite Series
Trang 2Section 8.3 Power Series
Trang 3Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 8.3, Slide 3
Up to this point we have dealt with infinite series whose terms were fixed numbers
We broaden our perspective now to consider series whose terms are variables
The simplest kind is known as a power series, and the main question will be determining the set of values of the variable for which the series is convergent
Let be a sequence of real numbers The series
is called a power series The number a n is called the nth coefficient of the series
Consider the power series whose coefficients are all equal to 1: xn
This is the geometric series, and it converges iff |x| < 1
Definition 8.3.1
Example 8.3.2
0
n n n
a n n0
Trang 4Let a n x n be a power series and let = lim sup |a n |1/n Define R by
Then the series converges absolutely whenever |x | < R and diverges whenever | x | > R
(When R = +, we take this to mean that the series converges absolutely for all real x
When R = 0, then the series converges only at x = 0.)
Proof:
Let b n = a n x n and apply the root test (Theorem 8.2.8) If lim sup |a n |1/n = , we have
= lim sup |b n |1/n = lim sup |a n x n |1/n = |x | lim sup |a n |1/n = |x |
Thus if = 0, then = 0 < 1, and the series converges (absolutely) for all real x
If 0 < < +, then the series converges when |x | < 1 and diverges when |x | > 1 That is, a n x n converges when |x | < 1/ = R and diverges when |x | > 1/ = R
Theorem 8.3.3
, if 0 , if 0,
R
< <
Trang 5Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 8.3, Slide 5
(Ratio Criterion)
The radius of convergence R of a power series a n x n is equal to limn |a n /a n + 1|,
provided that this limit exists
From Theorem 3 we see that the set of values C for which a power series converges
will either be {0}, , or a bounded interval centered at 0
The R that is obtained in the theorem is referred to as the radius of convergence
and the set C is called the interval of convergence
We think of {0} as an interval of zero radius and as an interval of infinite radius
When R = +, we may denote the interval of convergence by ( , )
Notice that when R is a positive real number, the theorem says nothing about the
convergence or divergence of the series at the endpoints of the interval of convergence
It is usually necessary to check the endpoints individually for convergence using
one of the other tests in Section 8.2
The following Ratio Criterion is also useful in determining the radius of convergence
Note that this ratio is the reciprocal of the ratio in the ratio test for sequences (Theorem 8.2.7.)
Theorem 8.3.4
Trang 6(a) The interval of convergence for the geometric series xn is (–1, 1).
(b) For the series we have
so the radius of convergence is 1
When x = 1, we get the divergent harmonic series:
When x = –1, we get the convergent alternating harmonic series:
So the interval of convergence is [–1, 1)
(c) For the series we have
so the radius of convergence is 1
When x = 1, we get the convergent p-series with p = 2:
When x = –1, the series is also absolutely convergent since
Example 8.3.5
1 n
n x
1
1
n
n
a
n
n
2
1 n
n x
1
( 1)
n
2
n
n
Trang 7Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 8.3, Slide 7
(a) For the series we have
Thus the radius of convergence is + , and the interval of convergence is (b) For the series , it is easier to use the root formula:
Thus R = 0 and the interval of convergence is {0}.
Consider the series
Letting y = x2, we may apply the ratio criterion to the series and obtain
Since it also converges when y = 3
but diverges when y = 3, its interval of convergence is [3,3)
Thus the series in y converges when | y | < 3
But y = x2, so the original series has as its interval of convergence
Note that is not included, because this corresponds to y = +3.
Example 8.3.7
Example 8.3.9
1
( 1)!
!
n n
1
! n
n x
n n
n x
1/
1/
lim a n n lim n n n
1
n n n
n
1(3 / )n n
1 1
3 ( 1)
3 ( )
n n
n n
3, 3
3
x
Trang 8Sometimes we wish to consider more general power series of the form
where x0 is a fixed real number
By making the substitution y = x – x0, we can apply the familiar techniques to the series
If we find that the series in y converges when | y | < R, we conclude that the original series
converges when |x – x0| < R.
For the series
we have R = 1, so it converges when | x – 1| < 1.That is, when 0 < x < 2.
Since it diverges at x = 0 and x = 2, the interval of convergence is (0,2)
Example 8.3.10
0 0
( ) ,n
n n
a x x
0
n n n
a y
0
n