4. Sequences
A sequence can be thought as a list of numbers
written in a definite order
,,,,,, 4321 naaaaa
{ }na
5. 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
6. Limit of a sequence (Definition)
A sequence has the limit if for every
there is a corresponding integer N such that
We write
{ }na L
∞→→=
∞→
nasLaorLa nn
n
lim
0>ε
NnwheneverLan ><− ,ε
7. 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
n
n
a
∞→
lim
8. 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
n
alim
NnwheneverMan >> ,
−∞=
∞→
n
n
alim
NnwheneverMan >−< ,
9. The limit laws
If and are convergent sequences and c is a
constant, then
{ }na { }nb
( )
( ) ccacac
baba
n
n
n
n
n
n
n
n
n
nn
n
=⋅=⋅
±=±
∞→∞→∞→
∞→∞→∞→
lim,limlim
limlimlim
10. The limit laws
( ) ( ) ( )
( ) ( ) 00,limlim
0lim,
lim
lim
lim
limlimlim
>>=
≠=
⋅=⋅
∞→∞→
∞→
∞→
∞→
∞→
∞→∞→∞→
n
p
n
n
p
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
nn
n
aandpifaa
bif
b
a
b
a
baba
11. 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
n
n aaaaa 321
1
12. 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
++++++=
++++++=
∑
∑
∞
=
∞
=
n
n
n
n
n
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
432
1
13. 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
n
nN aaaaaS ++++== ∑=
321
1
14. 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.
th
N NS
∑
∞
=1n
na S SSN
N
=
∞→
lim
∑
∞
=
=
1n
naS
∑
∞
=1n
na
15. Laws of Series
If and both converge, then
Note that the laws do not apply to multiplication,
division nor exponentiation.
∑
∞
=1n
na
∑
∞
=1n
nb
( )
∑∑
∑∑∑
∞
=
∞
=
∞
=
∞
=
∞
=
⋅=⋅
±=±
11
111
n
n
n
n
n
nn
n
n
n
n
acac
baba
16. 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 ∑
∞
=1n
na
( ) ∑∑∑∑∑
∞
=
∞
=
∞
=
∞
=
∞
=
−
11
2
111
sin
111
1
nnnnn
n
n
nnn
17. Proof Divergence Test
If , then
( ) 1
1
1321
1321
−
−
−
−
−=⇒
+=
+++++=
+++++=
nnn
nnn
nnn
nnn
SSa
aSS
aaaaaS
aaaaaS
∑
∞
=
=
1n
n Sa
18. Geometric Series
+⋅+⋅+⋅+⋅+=⋅∑
∞
=
432
0
rcrcrcrccrc
n
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
19. Geometric Series
If we multiply both sides by r we get
If we subtract (2) from (1), we get
)1(32
0
N
N
N
n
n
N
rcrcrcrccS
rcS
⋅++⋅+⋅+⋅+=
⋅= ∑=
)2(1432 +
⋅++⋅+⋅+⋅+⋅=⋅ N
N rcrcrcrcrcSr
( ) ( )
( )
r
rc
S
rcrS
rccSrS N
NN
N
N
NN
−
−
=⇒
−=−
⋅−=⋅− +
+
+
1
1
11
1
1
1
20. Geometric Series
An infinite GS diverges if , otherwise1≥r
1,
1
1
1,
1
1,
10
<
−
=⋅
<
−
⋅
=⋅
<
−
=⋅
∑
∑
∑
∞
=
∞
=
∞
=
r
r
term
rc
r
r
rc
rc
r
r
c
rc
st
Mn
n
M
Mn
n
n
n
21. Examples
( ) ( ) ( )
( )
∑∑
∑∑∑
∑∑∑∑
∞
=
∞
=
∞
=
∞
=
−
∞
=
∞
=
∞
=
−
∞
=
∞
=
+
+
−
−⋅
10
11
1
1
2000
52
ln
6
23
26.0
5
1
3
1
2113
nn
n
nn
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
22. P-Series
A p-series is a series of the form
Convergence of p-series:
++++=∑
∞
=
ppp
n
p
n 4
1
3
1
2
1
1
11
1
≤
>
=∑
∞
= 1
11
1 pforDiverges
pforConverges
nn
p
24. Comparison Test
Assume that there exists such that
for
1. 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
0>M nn ba ≤≤0
Mn ≥
∑
∞
=1n
nb ∑
∞
=1n
na
∑
∞
=1n
na ∑
∞
=1n
nb
∑
∞
=1n
nb
∑
∞
=1n
na
25. 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
a
L
∞→
= lim
0>L ∑
∞
=1n
na
∑
∞
=1n
nb0=L ∑
∞
=1n
nb
∑
∞
=1n
na
27. Absolute/Conditional Convergence
is called absolutely convergent if
converges
Absolute convergence theorem:
If convs. Also convs.
(In words) if convs. Abs. convs.
∑
∞
=1n
na ∑
∞
=1n
na
∑
∞
=1n
na ⇒∑
∞
=1n
na
∑
∞
=1n
na ⇒∑
∞
=1n
na
( )
∑∑
∞
=
∞
=
−
−
1
2
1
1
2
1
n
n
n
n
n
28. 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 1
lim +
∞→
=ρ
1<ρ ∑
∞
=1n
na
1=ρ
1>ρ
∑
∞
=1n
na
( ) ∑∑∑∑∑
∞
=
−
∞
=
∞
=
∞
=
∞
=
−
1
2
1
2
11
2
1 100
!
1
2!
1
nnn
n
n
n
n
n
nn
nn
n
Examples
29. 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
n
n
n
aL
∞→
= lim
1<L ∑
∞
=1n
na
1=L
1>L ∑
∞
=1n
na
∑∑∑∑
∞
=
−
∞
=
∞
=
∞
=
+ 1
2
1
2
1
2
1 232 nnn
n
n
n
nn
n
n
n
Examples
30. Power Series
A power series is a series of the form:
( ) ( ) ( )
+−+−+=−
+++++=
∑
∑
∞
=
∞
=
2
21
0
0
2
21
0
0
axcaxccaxc
xcxcxccxc
n
n
n
n
n
n
n
n
31. Power Series
Theorem: For a given power series
there are 3 possibilities:
1. The series converges only when
2.The series converges for all
3. There is a positive number R, such that the series
converges if and diverges if
( )∑
∞
=
−
0n
n
n axc
ax =
x
Rax <− Rax >−
32. Taylor & Maclaurin Series
Let , then
therefore ,
,234)0(,23)0(,2)0(,)0(,)0( 4
)(
3210 afafafafaf IV
⋅⋅=⋅=′′′=′′=′=
+++++== ∑
∞
=
4
4
3
3
2
210
0
)( xaxaxaxaaxaxf n
n
n
n
n
nk
k x
n
f
xf
k
f
a ∑
∞
=
=⇒=
0
)()(
!
)0(
)(
!
)0(
2
2
cossin xxx
eexxxe
Examples
−