Ch.2 Limits and derivatives
Download
Report
Transcript Ch.2 Limits and derivatives
The comparison tests
Theorem Suppose that an and bn are series with
positive terms, then
(i) If bn is convergent and an bn for all n, then an is
also convergent.
(ii) If bn is divergent and an bn for all n, then an is also
divergent.
1
Ex. Determine whether n
converges.
n 1 2 1
1
1
n
Sol.
So the series converges.
n
2 1 2
The limit comparison test
Theorem Suppose that an and bn are series with
positive terms. Suppose
an
lim
c.
n b
n
Then
(i) when c is a finite number and c>0, then either both series
converge or both diverge.
(ii) when c=0, then the convergence of bn implies the
convergence of an .
(iii) when c , then the divergence of bn implies the
divergence of an .
Example
Ex. Determine whether the following series converges.
1
2n 2 3n
p
(3) sin
(2) 2
(1)
n
n 1
n 1 ln ( n 1)
n 1
5 n5
Sol. (1) diverge. choose bn 1/ n1/ 2 then lim an / bn 2
n
(2) diverge. take bn 1/ n
an / bn
then lim
n
(3) converge for p>1 and diverge for p 1 take bn 1/ n p
p
then lim an / bn
n
Question
Ex. Determine whether the series
converges or diverges.
Sol. an a
ln
1
n
e
1
ln ln a
n
1
ln a
n
diverge for 0 a e
converge for a e
a
n 1
ln
1
n
(a 0)
Alternating series
An alternating series is a series whose terms are
alternatively positive and negative. For example,
1 1 1
(1)n1
1
2 3 4
n
n 1
The n-th term of an alternating series is of the form
an (1) n 1 bn or an (1) n bn
where bn is a positive number.
The alternating series test
Theorem If the alternating series
n 1
(
1)
bn b1 b2 b3 b4 b5 b6
n 1
(bn 0)
bn 0
satisfies (i) bn 1 bn for all n (ii) lim
n
Then the alternating series is convergent.
(1)n1
Ex. The alternating harmonic series
n
n 1
is convergent.
Example
Ex. Determine whether the following series converges.
(1)n1
(1)n1 n2
(1)
( 0)
(2) 3
n
n 1
n 1
n 1
Sol. (1) converge
(1)n1 n
Question.
n 1 4n 1
(2) converge
Absolute convergence
A series an is called absolutely convergent if the series
of absolute values | an | is convergent.
(1)n1
For example, the series 3/ 2 is absolutely convergent
n 1 n
while the alternating harmonic series is not.
A series an is called conditionally convergent if it is
convergent but not absolutely convergent.
Theorem. If a series is absolutely convergent, then it is
convergent.
Example
Ex. Determine whether the following series is convergent.
sin n
(1) 2
n 1 n
(1)n
(2)
n 1 ln(1 n)
Sol. (1) absolutely convergent
(2) conditionally convergent
The ratio test
The ratio test
an 1
(1) If lim
L 1, then an is absolutely convergent.
n a
n 1
n
an 1
an 1
(2) If lim
then an diverges.
L 1 or lim
n a
n a
n 1
n
n
an1
(3) If lim
1, the ratio test is inconclusive: that is, no
n a
n
conclusion can be drawn about the convergence of
a
n 1
n
Example
Ex. Test the convergence of the series
an
(1)
n 1 n !
an n!
(2) n
n 1 n
Sol. (1) convergent
(2) convergent for a e; divergent for a e
an 1 a n 1 (n 1)! a n n !
a
a
/ n
n 1
n
an
(n 1)
n
(1 1/ n)
e
a e an 1 an lim an 0
n
The root test
The root test
(1) If lim n | an | L 1, then an is absolutely convergent.
n
n 1
(2) If lim n | an | L 1 or lim n | an | then an diverges.
n
n
n 1
(3) If lim n | an | 1, the root test is inconclusive.
n
Example
Ex. Test the convergence of the series
n 1
n
1
a
n
n
n
1
an lim
Sol. lim
n
n
1 a
a
n
convergent for a 1; divergent for 0 a 1
n
n
n
a 1 an
(n )
1 n
(1 )
n
(a 0)
Rearrangements
If we rearrange the order of the term in a finite sum, then
of course the value of the sum remains unchanged. But this
is not the case for an infinite series.
By a rearrangement of an infinite series an we mean a
series obtained by simply changing the order of the terms.
It turns out that: if an is an absolutely convergent series
with sum s , then any rearrangement of an has the
same sum s .
However, any conditionally convergent series can be
rearranged to give a different sum.
Example
Ex. Consider the alternating harmonic series
1 1 1 1 1
1 ln 2.
2 3 4 5 6
Multiplying this series by 1/ 2, we get
1 1 1 1 1 1
1
ln 2.
2 4 6 8 10 12
2
1
1
1
1
1
or
0 0 0 0 ln 2.
2
4
6
8
2
Adding these two series, we obtain
1 1 1 1 1
3
1 ln 2.
3 2 5 7 4
2
Strategy for testing series
If we can see at a glance that lim an 0 then divergence
n
n 1
1
an
0
2n 1
2
If a series is similar to a p-series, such as an algebraic form,
or a form containing factorial, then use comparison test.
n3 1
1
an 3
~ 3/ 2
2
3n 4n 2 3n
For an alternating series, use alternating series test.
3
n
an (1)n 4
n 1
Strategy for testing series
If n-th powers appear in the series, use root test.
n2
an ne
If an f (n), f decreasing and positive, use integral test.
1
an
n(ln n)(ln ln n)
1
(1)n ln n
2n n !
1
(1) tan
(2)
(3)
(4)
n
(n 2)!
(ln n)ln n
n
Sol. (1) diverge (2) converge (3) diverge (4) converge
Power series
A power series is a series of the form
n
2
3
c
x
c
c
x
c
x
c
x
n
0
1
2
3
n 0
where x is a variable and cn are constants called coefficients
of series.
For each fixed x, the power series is a usual series. We can
test for convergence or divergence.
A power series may converge for some values of x and
diverge for other values of x. So the sum of the series is a
function s ( x ) c c x c x 2 c x 3
0
1
2
3
Power series
For example, the power series
n
2
3
x
1
x
x
x
n 0
1
converges to s ( x)
when 1 x 1.
1 x
More generally, A series of the form
n
2
c
(
x
a
)
c
c
(
x
a
)
c
(
x
a
)
n
0
1
2
n 0
is called a power series in (x-a) or a power series centered
at a or a power series about a.
Example
Ex. For what values of x is the power series
convergent?
Sol. By ratio test,
n
n
!
x
n 0
an 1
(n 1)! x n 1
lim
lim
lim(n 1) | x |
n
n a
n
n
n! x
n
the power series diverges for all x 0, and only converges
when x=0.
Homework 24
Section 11.4: 24, 31, 32, 42, 46
Section 11.5: 14, 34
Section 11.6: 5, 13, 23
Section 11.7: 7, 8, 10, 15, 36