74
CALCULUS II Chapter 10
| Date post: | 03-Jan-2016 |
| Category: |
Documents |
| Upload: | rhoslyn-bronwen |
| View: | 34 times |
| Download: | 0 times |
CALCULUS II
Chapter 10
10.1 Sequences
• A sequence can be thought as a list of numbers written in a definite order
,,,,,, 4321 naaaaa
na
Examples
,3,,3,2,1,03
,3
11,,
27
4,9
3,3
2
3
11
,1
,,4
3,3
2,2
1
1
3
nn
nn
n
n
n
n
n
n
n
n
n
http://www.youtube.com/watch?v=Kxh7yJC9Jr0
Limit of a sequence
• Consider the sequence• If we plot some values we get this graph
1n
n
Limit of a sequence
• Consider the sequence
12
1
nn
n
Limit of a sequence
• Since a sequence is a collection of numbers, we could have a random collection
Limit of a sequence
• Consider the Fibonacci sequence
1,1, 2111 aaaaa nnn
Limit of a sequence
Limit of a sequence (Definition 1)
• A sequence has the limit if we can make the terms as close as we like by taking n sufficiently large.
• We write
na Lna
nasLaorLa nnnlim
Limit of a sequence (Definition 2)
• A sequence has the limit if for every there is a corresponding integer N such
that
• We write
na L
nasLaorLa nnnlim
0
NnwheneverLan ,
Convergence/Divergence
• If exists we say that the sequence converges.– Note that for the sequence to converge, the limit
must be finite
• If the sequence does not converge we will say that it diverges– Note that a sequence diverges if it approaches to
infinity or if the sequence does not approach to anything
nna
lim
Divergence to infinity
• means that for every positive number M there is an integer N such that
• means that for every positive number M there is an integer N such that
n
nalim
NnwheneverMan ,
n
nalim
NnwheneverMan ,
The limit laws
• If and are convergent sequences and c is a constant, then na nb
ccacac
baba
nn
nn
n
nn
nn
nnn
lim,limlim
limlimlim
The limit laws
00,limlim
0lim,lim
limlim
limlimlim
np
nn
pn
n
nn
nn
nn
n
n
n
nn
nn
nnn
aandpifaa
bifb
a
b
a
baba
L’Hopital and sequences
• Theorem: If and , when n is an integer, then
• L’Hopital: Suppose that and are differentiable and that near a. Also suppose that we have an indeterminate form of type . Then
Lxfx
)(lim nanf )(Lan
n
lim
)(xf )(xg0)(' xg
or0
0
)(
)(lim
)(
)(lim
xg
xf
xg
xfaxax
More Theorems
• Squeeze thm: Let be sequences such that for some M, for and . Then
• Continuity: If is continuous and the limit exists, then
• Bounded monotonic sequences converge: if for all n, and
nnn cba ,,
nnn cab Mn Lcb n
nn
n
limlim Lan
n
lim
)(xf
Lann
lim )()lim()(lim Lfafaf nn
nn
1 nn aa Man
Examples
!
10limlim
lnlimlim
2
1lim
8
1lim7lim5lim
1limlim!
lim4
220lim
limln
lim1
lim1
4lim
2
1
22
1
1
3
2
2
2
ne
n
n
nn
n
n
nnen
n
n
nn
n
n
n
n
n
n
n
n
nnnn
n
n
n
n
n
n
n
n
n
n
n
n
nnnn
nnnn
http://www.youtube.com/watch?v=9K1xx6wfN-U
10.2 Infinite Series
• Is the summation of all elements in a sequence.
• Remember the difference: Sequence is a collection of numbers, a Series is its summation.
n
nn aaaaa 321
1
http://www.youtube.com/watch?v=haK3oC0L_a8
Visual proof of convergence
• It seems difficult to understand how it is possible that a sum of infinite numbers could be finite. Let’s see an example
nn
n
nn
n
2
1
16
1
8
1
4
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
4321
12
1
16
1
8
1
4
1
2
1
2
1
1
n
nn
Convergence/Divergence
• We say that an infinite series converges if the sum is finite, otherwise we will say that it diverges.
• To define properly the concepts of convergence and divergence, we need to introduce the concept of partial sum
N
N
nnN aaaaaS
3211
Convergence/Divergence
• The partial sum is the finite sum of the first terms.
• converges to if and we write:
• If the sequence of partial sums diverges, we say that diverges.
thN NSN
1nna S SSN
N
lim
1n
naS
1nna
Laws of Series
• If and both converge, then
– Note that the laws do not apply to multiplication, division nor exponentiation.
1nna
1nnb
11
111
nn
nn
nnn
nn
nn
acac
baba
Divergence Test
• If does not converge to zero, then diverges.– Note that in many cases we will have sequences
that converge to zero but its sum diverges
na
1nna
11
2111
sin111
1nnnnn
n nnnn
Proof Divergence Test
• If , then
1
1
1321
1321
nnn
nnn
nnn
nnn
SSa
aSS
aaaaaS
aaaaaS
1n
n Sa
n
nalim
Geometric Series
432
0
rcrcrcrccrcn
n
Note that in this case we start counting from zero. Technically it doesn’t matter, but we have to be careful because the formula we will use starts always at n=0.
First term multiplied by r
Second term multiplied by r
Third term multiplied by r
Geometric Series
If we multiply both sides by r we get
If we subtract (2) from (1), we get
)1(32
0
NN
N
n
nN
rcrcrcrccS
rcS
)2(1432 NN rcrcrcrcrcSr
r
rcS
rcrS
rccSrS N
NNN
NNN
1
1
11
1
1
1
Geometric Series
• An infinite GS diverges if , otherwise 1r
1,1
1
1,1
1,10
rr
termrc
rr
rcrc
rr
crc
st
Mn
n
M
Mn
n
n
n
Examples (not only GS)
10
11
1
1
2000
52ln
6
23
26.05
1
3
12113
nnn
nn
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
http://www.youtube.com/watch?v=xjmy5hkZccY
http://www.youtube.com/watch?v=C8piSCOdo1Y
Telescoping Series
To solve we will use the identity:
133
1
122
1
111
1
1
1
1n nn
1
11
1
1
nnnn
Telescoping Series
11
11limlim
1
11
11
1
11
4
1
3
1
3
1
2
1
2
1
1
1
1
11
1
1
1
1 1
NS
nn
NS
NNS
nnnnS
NN
Nn
N
N
N
n
N
nN
http://www.youtube.com/watch?v=7tDK_UjdWOs
http://www.youtube.com/watch?v=MDYb5DnRH2c
Harmonic Series
• Basically this implies that
TOO BIG!!!
1
1
n n
http://www.youtube.com/watch?v=0XIqnoJ72CU
P-Series
• A p-series is a series of the form
• Convergence of p-series:
ppp
npn 4
1
3
1
2
1
1
11
1
1
11
1 pforDiverges
pforConverges
nnp
Examples (not only P-series)
111
1
1
1
11
5
1
1
5
1
115
1
5
1
5
1
5
153
11
1
11
12
2
1
001.0
112
5
1
5
1555
51
11
5
6
3
45ln
11
nn
nn
n
n
n
n
n
n
n
nn
n
nnnn
nnn
n
nn
n
nnnn
n
nn
nnn
nn
n
n
n
n
nn
Comparison Test
• Assume that there exists such that for1. If converges, then also converges.
2. If diverges, then also diverges.
– if diverges this test does not help – Also, if converges this test does not help
0M nn ba 0Mn
1nnb
1nna
1nna
1nnb
1nnb
1nna
Limit Comparison Test
• Let and be positive sequences. Assume that the following limit exists
• If , then converges if and only if converges. (Note that L can not be infinity)
• If and converges, then converges
na nb
n
n
n b
aL
lim
0L
1nna
1nnb
0L
1nnb
1nna
Examples
111
13
2
11
3
1
12
14
2
11
2
4ln
4
1
ln
4
11
4
1
3
12
n
n
nn
nnn
nnnn
n
n
ennn
n
n
n
nn
n
nn
nnn
n
n
http://www.youtube.com/watch?v=xesQnFWw8f8
http://www.youtube.com/watch?v=8eCFY82HkRA
Absolute/Conditional Convergence
• is called absolutely convergent if converges
• Absolute convergence theorem: – If convs. Also convs.– (In words) if convs. Abs. convs.
1nna
1nna
1nna
1nna
1nna
1nna
12
1
1
2
1
n
n
n
n
n
http://www.youtube.com/watch?v=6hOeqjoQvNw
Leibniz Test for alternating series
• Let be:– Decreasing – Positive – Converging to 0Then,
Converges
na
0na
0nann aa 1
1
1n
nna
1
1
1
1
1
n
n
n
n
n
n
Examples
Ratio Test
• Let be a sequence and assume that the following limit exists:
– If , then converges absolutely– If , then diverges– If , the Ratio Test is INCONCLUSIVE
na
n
n
n a
a 1lim
1
1nna
1
1
1nna
1
2
1
2
11
2
1 100
!1
2!
1
nnnn
n
nn
n
nnnn
n
Examples
http://www.youtube.com/watch?v=iy8mhbZTY7g
http://www.youtube.com/watch?v=iy8mhbZTY7g&feature=PlayList&p=21CAE1F783F4C165&index=3
http://www.youtube.com/watch?v=gay5CEXnkhA
http://www.youtube.com/watch?v=iy8mhbZTY7g&feature=PlayList&p=21CAE1F783F4C165&index=3
Root Test
• Let be a sequence and assume that the following limit exists:
– If , then converges absolutely– If , then diverges– If , the Ratio Test is INCONCLUSIVE
na
nn
naL
lim
1L
1nna
1L
1L
1nna
1
2
1
2
1
2
1 232 nnnn
n
n
nnn
n
n
Examples
http://www.youtube.com/watch?v=vDdDLfIya0I
http://www.youtube.com/watch?v=iy8mhbZTY7g&feature=PlayList&p=21CAE1F783F4C165&index=3
Examples
• Use this strategy to test the series in the following exams:
• Exam 1• Exam 2• Exam 3• Exam 4
http://www.youtube.com/watch?v=DvadVYHf3pM
http://www.youtube.com/watch?v=iy8mhbZTY7g&feature=PlayList&p=21CAE1F783F4C165&index=3
Power Series
• A power series is a series of the form:
221
00
221
00
axcaxccaxc
xcxcxccxc
n
nn
nn
n
nn
Power Series
• Theorem: For a given power series there are 3 possibilities:
1.The series converges only when2.The series converges for all3.There is a positive number R, such that the
series converges if and diverges if
0n
nn axc
ax x
Rax
Rax
http://www.youtube.com/watch?v=01LzAU__J-0&feature=channel
http://www.youtube.com/watch?v=MM3BXtVu9eM&feature=response_watch
Examples
• For what values of x do the following power series converge?
05
0 032
001
0
022
2
00
5
1
1
414
ln4
1
3
2
1
3
!2
13!
nn
nn
n n
nn
nn
nn
nn
n
n
nn
nn
nn
n
n
n
n
n
x
n
xn
n
x
n
xxn
n
x
n
x
n
xxn
Representation of fns as PS
• From the Geometric Series formula we can deduce:
• Theorem: Term-by-term differentiation, if converges for , thenis its derivative and also converges for
11
1
0
n
n xforx
x
0n
nnxa Rx
0
1
n
nn xna
Rx
4322 54
1
23
1
5
1
1
1
31
1
xxxxx
http://www.youtube.com/watch?v=XWGPjZK0Yzw&feature=related
Representation of fns as PS
• Theorem: Term-by-term differentiation, if converges for , thenis its derivative and also converges for
• Theorem: Term-by-term anti-differentiation, if converges for , thenis its derivative and also converges for
0n
nnxa Rx
0
1
n
nn xna
Rx
0n
nnxa Rx
0
1
1n
n
n n
xa
Rx
xx
xx41ln1ln
21
1
1
132
Taylor & Maclaurin Series
• Let , then
therefore ,,234)0(,23)0(,2)0(,)0(,)0( 4
)(3210 afafafafaf IV
44
33
2210
0
)( xaxaxaxaaxaxf n
nn
n
n
nk
k xn
fxf
k
fa
0
)()(
!
)0()(
!
)0(
22cossin xxx eexxxe
Examples
http://www.youtube.com/watch?v=cjPoEZ0I5wQ&feature=related
http://www.youtube.com/watch?v=Os8OtXFBLkY&feature=related
http://www.youtube.com/watch?v=qPl9nr8my2Q&feature=related
The Binomial Series
In general
terefore
21)0(121)(
1)0(11)(
)0(1)(
1)0(1)(
3
2
1
aaafxaaaxf
aafxaaxf
afxaxf
fxxf
a
a
a
a
121)0()( naaaaf n
!
121
!
)0()(
n
naaaa
n
f n
31
2
3
1
2
1
1
1
1
:
x
x
x
Examples
http://www.youtube.com/watch?v=ocNLs4hmNtI
http://www.youtube.com/watch?v=Jgy-MLHlzO0&feature=related
