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Shantilal Shah Engineering College, Bhavnagar.
Mechanical Engineering Department
SequencesA 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
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 nnnlim
0
NnwheneverLan ,
Convergence/DivergenceIf exists we say that the sequence
converges.Note that for the sequence to converge, the
limit must be finiteIf the sequence does not converge we will
say that it divergesNote 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 lawsIf 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
Infinite SeriesIs 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
Visual proof of convergenceIt 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
Convergence/DivergenceWe 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/DivergenceThe 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 NS
1nna S SSN
N
lim
1n
naS
1nna
Laws of SeriesIf 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 TestIf 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
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 SeriesAn 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
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
P-SeriesA 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
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 TestAssume 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
0Mnn ba 0
Mn
1nnb
1nna
1nna
1nnb
1nnb
1nna
Limit Comparison TestLet 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
1nnb0L
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
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
Ratio TestLet be a sequence and assume that the
following limit exists:
If , then converges absolutely If , then divergesIf , 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
Root TestLet be a sequence and assume that the
following limit exists:
If , then converges absolutely If , then divergesIf , 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
Power SeriesA power series is a series of the form:
221
00
221
00
axcaxccaxc
xcxcxccxc
n
nn
nn
n
nn
Power SeriesTheorem: 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
Taylor & Maclaurin SeriesLet ,
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
