Sequences n series Civil Engineering (CE) Notes | EduRev

Civil Engineering (CE) : Sequences n series Civil Engineering (CE) Notes | EduRev

 Page 1


chap-02 B.V.Ramana August 30, 2006 10:15
Chapter2
Sequences and Series
INTRODUCTION
The study of convergence and divergence of a se-
quence, which is an ordered list of things, is a prereq-
uisit for in?nite series. The unit square in the ?gure
can be expressed as an in?nite (geometric) series
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+···
Several functions can be expressed as “in?nite
polynomials” (known as “powerser ies”) using
the concept of in?nite series. By Fourier series,
certain functions can be represented as an in?nite
sum of trigonometric functions. Using in?nite
series, differential equations in problems of signal
transmission, chemical diffusion, vibration and heat
?ow can be solved and non elementary integrals
evaluated. The in?nite process of summing of an
in?nite series is a puzzle for centuries convergence
and divergence of in?nite series plays an important
role in engineering applications.
1
2
1
4
1
8
1
16
2.1 SEQUENCES
A sequence is a function from the domain set of
natural numbersN to any setS.
Real sequence isafunctionfromN to R,
the set of real numbers; denoted by f :N ? R.
Thus the real sequence f is set of all ordered
pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs
(n,f (n)) with n a positive integer.
Notation: Sincethedomainofasequenceisalways
thesame(thesetofpositiveintegers)asequencemay
be written as{f (n)} instead of{n,f (n)}.
Examples:
1.

n,
1
n

=

1
n

=

1,
1
2
,
1
3
,
1
4
,...
1
n
...

2.

n,
1
2
n-1

=

1
2
n-1

=

1,
1
2
,
1
2
2
,
1
2
3
,
1
2
4
,...,
1
2
n-1
,...

constant sequence where range is singleton set{c},
c= constant.
Example: {3, 3, 3, 3,...}
Null sequence{0, 0, 0,..., 0,...}
A sequence is also denoted by{a
n
} whose ordinate
y = a
n
at the abscissax = n. Thus in a sequence for
each positive integern, a numbera
n
is assigned and
is denoted as	a
n

or( a
n
)or
{a
n
}={a(1),a(2),a(3),...,a(n),...}
={a
1
,a
2
,a
3
,...,a
n
,...}
Herea
1
,a
2
,a
3
,...a
n
, are known as the ?rst, second,
third and nth terms of the sequence.
In?nite sequence is a sequence in which the
numberoftermsisin?nite,andisdenotedby {a
n
}
8
n=1
.
Ontheotherhand,?nitesequencedenotedby{a
n
}
m
n=1
containsonlya?nitenumberofterms( m=?nite).
Bounded sequence A sequence {a
n
} is said to
be bounded if there exists numbers m and M such
thatm<a
n
<M forevery n, otherwise it is said to
be unbounded.
Monotonic sequence
A sequence{a
n
}issaidtobe
a. monotonically increasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
2.1
Page 2


chap-02 B.V.Ramana August 30, 2006 10:15
Chapter2
Sequences and Series
INTRODUCTION
The study of convergence and divergence of a se-
quence, which is an ordered list of things, is a prereq-
uisit for in?nite series. The unit square in the ?gure
can be expressed as an in?nite (geometric) series
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+···
Several functions can be expressed as “in?nite
polynomials” (known as “powerser ies”) using
the concept of in?nite series. By Fourier series,
certain functions can be represented as an in?nite
sum of trigonometric functions. Using in?nite
series, differential equations in problems of signal
transmission, chemical diffusion, vibration and heat
?ow can be solved and non elementary integrals
evaluated. The in?nite process of summing of an
in?nite series is a puzzle for centuries convergence
and divergence of in?nite series plays an important
role in engineering applications.
1
2
1
4
1
8
1
16
2.1 SEQUENCES
A sequence is a function from the domain set of
natural numbersN to any setS.
Real sequence isafunctionfromN to R,
the set of real numbers; denoted by f :N ? R.
Thus the real sequence f is set of all ordered
pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs
(n,f (n)) with n a positive integer.
Notation: Sincethedomainofasequenceisalways
thesame(thesetofpositiveintegers)asequencemay
be written as{f (n)} instead of{n,f (n)}.
Examples:
1.

n,
1
n

=

1
n

=

1,
1
2
,
1
3
,
1
4
,...
1
n
...

2.

n,
1
2
n-1

=

1
2
n-1

=

1,
1
2
,
1
2
2
,
1
2
3
,
1
2
4
,...,
1
2
n-1
,...

constant sequence where range is singleton set{c},
c= constant.
Example: {3, 3, 3, 3,...}
Null sequence{0, 0, 0,..., 0,...}
A sequence is also denoted by{a
n
} whose ordinate
y = a
n
at the abscissax = n. Thus in a sequence for
each positive integern, a numbera
n
is assigned and
is denoted as	a
n

or( a
n
)or
{a
n
}={a(1),a(2),a(3),...,a(n),...}
={a
1
,a
2
,a
3
,...,a
n
,...}
Herea
1
,a
2
,a
3
,...a
n
, are known as the ?rst, second,
third and nth terms of the sequence.
In?nite sequence is a sequence in which the
numberoftermsisin?nite,andisdenotedby {a
n
}
8
n=1
.
Ontheotherhand,?nitesequencedenotedby{a
n
}
m
n=1
containsonlya?nitenumberofterms( m=?nite).
Bounded sequence A sequence {a
n
} is said to
be bounded if there exists numbers m and M such
thatm<a
n
<M forevery n, otherwise it is said to
be unbounded.
Monotonic sequence
A sequence{a
n
}issaidtobe
a. monotonically increasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
2.1
chap-02 B.V.Ramana August 30, 2006 10:15
2.2 ENGINEERING MATHEMATICS
b. monotonically decreasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
c.monotonic if it is eithermonotonically
increasing or monotonically decreasing.
Example:

1
n

=

1,
1
2
,
1
3
,
1
4
,
1
5
,...

bounded
since 0<a
n
=
1
n
< 1 and monotonically decreasing.
Example: {2
n
}={2, 2
2
, 2
3
, 2
4
,...} unbounded
since 2
n
becomes larger and larger asn comes large
and monotonically increasing.
2.2 LIMIT OF A SEQUENCE
Considerasequence {a
n
}=

3+
1
n

.
Plotting the values
n: 1 2 4 5 10 50 100 1000 10000 100000...
a
n
: 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001...
As n increases, a
n
= 3+
1
n
becomescloserto3.
Thus the difference (or distance) between 3+
1
n
and
3becomessmallerandsmalleras n becomes larger
and larger i.e., we can make 3+
1
3
and3asclose
as we please, by choosing an appropriately (suf?-
ciently) large value forn, i.e., the terms of a sequence
clusteraroundthis(limit)point.Howevernotethat
3+
1
n
=3foranyvalueof n.
Limit: A number L is said to be a limit of a
sequence{a
n
} and is denoted as
lim
As n?8
a
n
= lim
n?8
a
n
= lim a
n
= L
ifforevery > 0 there existsN such that
|a
n
-L|< forall n=N.
Note: Asequencemayhaveauniquelimitormay
havemorethanonelimitormaynothavealimitatall.
Result: A monotonic sequence always has a limit
(maybe?niteorin?nite).
2.3 CONVERGENCE, DIVERGENCE AND
OSCILLATION OF A SEQUENCE
Convergent A sequence{a
n
} is said to be conver-
gent if it has a ?nite limit i.e., lim
n?8
a
n
= L= ?nite
unique limit value.
Divergent If lim
n?8
a
n
= in?nite=±8.
Oscillatory If limit ofa
n
is not unique (oscillates
?nitely) or±8 (oscillates in?nitely).
Examples:
1.

1
n
2

convergent since lim
n?8
1
n
2
= 0=?nite unique
2. {n:} divergent since lim
n?8
n :=8
3. {(-1)
n
} oscillates ?nitely, since
lim
n?8
(-1)
n
=

1,n even
-1,n odd.
4. {(-1)
n
·n
2
} oscillates in?nitely,
since limit=±8.
Result1: If sequence{a
n
} converges to limitL and
{b
n
} converges toL
*
then
a. {a
n
+b
n
} converges toL+L
*
b. {ca
n
} converges toCL
c. {a
n
·b
n
} converges toL·L
*
d. {
a
n
b
n
} converges to
L
L
*
, providedL
*
= 0.
Result2: Every convergent sequence is bounded.
Example:

1
n

is convergent and is bounded
a
n
=
1
n
< 1,forevery n.
Result3: The converse is not true i.e., a bounded
sequence may not be convergent.
Example:{(-1)
n
} is oscillatory (has more than one
limit but is bounded since-1= (-1)
n
= 1.
Result4: A bounded monotonic sequence is con-
vergent.
Example:

1
n
2

is bounded since
1
n
2
=1forevery n
and monotonically decreasing since
1
n
2
>
1
(n+1)
2
for
every n. Hence the sequence is convergent because
lim
n?8
a
n
= lim
n?8
1
n
2
= 0= ?nite.
Useful Standard Limits
1. a. lim
n?8
1
n
= 0, b. lim
n?8
1
n
2
= 0, c. lim
n?8
1
v
n
= 0
2. lim
n?8
n
1/n
= 1
3. lim
n?8
logn
n
= 0
4. lim
n?8

1+
x
n

n
= e
x
, forany x
5. lim
n?8
x
1/n
=1forx> 0
6. (a) lim
n?8
x
n
=0for |x| <1i.e. - 1<x<1.
(b) lim
n?8
x
n
n!
=0 forany x. In formulas (5) and
6(b)x remains ?xed asn?8
WORKED OUT EXAMPLES
Determine the nature of the following sequences
whose nth term a
n
is
Page 3


chap-02 B.V.Ramana August 30, 2006 10:15
Chapter2
Sequences and Series
INTRODUCTION
The study of convergence and divergence of a se-
quence, which is an ordered list of things, is a prereq-
uisit for in?nite series. The unit square in the ?gure
can be expressed as an in?nite (geometric) series
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+···
Several functions can be expressed as “in?nite
polynomials” (known as “powerser ies”) using
the concept of in?nite series. By Fourier series,
certain functions can be represented as an in?nite
sum of trigonometric functions. Using in?nite
series, differential equations in problems of signal
transmission, chemical diffusion, vibration and heat
?ow can be solved and non elementary integrals
evaluated. The in?nite process of summing of an
in?nite series is a puzzle for centuries convergence
and divergence of in?nite series plays an important
role in engineering applications.
1
2
1
4
1
8
1
16
2.1 SEQUENCES
A sequence is a function from the domain set of
natural numbersN to any setS.
Real sequence isafunctionfromN to R,
the set of real numbers; denoted by f :N ? R.
Thus the real sequence f is set of all ordered
pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs
(n,f (n)) with n a positive integer.
Notation: Sincethedomainofasequenceisalways
thesame(thesetofpositiveintegers)asequencemay
be written as{f (n)} instead of{n,f (n)}.
Examples:
1.

n,
1
n

=

1
n

=

1,
1
2
,
1
3
,
1
4
,...
1
n
...

2.

n,
1
2
n-1

=

1
2
n-1

=

1,
1
2
,
1
2
2
,
1
2
3
,
1
2
4
,...,
1
2
n-1
,...

constant sequence where range is singleton set{c},
c= constant.
Example: {3, 3, 3, 3,...}
Null sequence{0, 0, 0,..., 0,...}
A sequence is also denoted by{a
n
} whose ordinate
y = a
n
at the abscissax = n. Thus in a sequence for
each positive integern, a numbera
n
is assigned and
is denoted as	a
n

or( a
n
)or
{a
n
}={a(1),a(2),a(3),...,a(n),...}
={a
1
,a
2
,a
3
,...,a
n
,...}
Herea
1
,a
2
,a
3
,...a
n
, are known as the ?rst, second,
third and nth terms of the sequence.
In?nite sequence is a sequence in which the
numberoftermsisin?nite,andisdenotedby {a
n
}
8
n=1
.
Ontheotherhand,?nitesequencedenotedby{a
n
}
m
n=1
containsonlya?nitenumberofterms( m=?nite).
Bounded sequence A sequence {a
n
} is said to
be bounded if there exists numbers m and M such
thatm<a
n
<M forevery n, otherwise it is said to
be unbounded.
Monotonic sequence
A sequence{a
n
}issaidtobe
a. monotonically increasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
2.1
chap-02 B.V.Ramana August 30, 2006 10:15
2.2 ENGINEERING MATHEMATICS
b. monotonically decreasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
c.monotonic if it is eithermonotonically
increasing or monotonically decreasing.
Example:

1
n

=

1,
1
2
,
1
3
,
1
4
,
1
5
,...

bounded
since 0<a
n
=
1
n
< 1 and monotonically decreasing.
Example: {2
n
}={2, 2
2
, 2
3
, 2
4
,...} unbounded
since 2
n
becomes larger and larger asn comes large
and monotonically increasing.
2.2 LIMIT OF A SEQUENCE
Considerasequence {a
n
}=

3+
1
n

.
Plotting the values
n: 1 2 4 5 10 50 100 1000 10000 100000...
a
n
: 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001...
As n increases, a
n
= 3+
1
n
becomescloserto3.
Thus the difference (or distance) between 3+
1
n
and
3becomessmallerandsmalleras n becomes larger
and larger i.e., we can make 3+
1
3
and3asclose
as we please, by choosing an appropriately (suf?-
ciently) large value forn, i.e., the terms of a sequence
clusteraroundthis(limit)point.Howevernotethat
3+
1
n
=3foranyvalueof n.
Limit: A number L is said to be a limit of a
sequence{a
n
} and is denoted as
lim
As n?8
a
n
= lim
n?8
a
n
= lim a
n
= L
ifforevery > 0 there existsN such that
|a
n
-L|< forall n=N.
Note: Asequencemayhaveauniquelimitormay
havemorethanonelimitormaynothavealimitatall.
Result: A monotonic sequence always has a limit
(maybe?niteorin?nite).
2.3 CONVERGENCE, DIVERGENCE AND
OSCILLATION OF A SEQUENCE
Convergent A sequence{a
n
} is said to be conver-
gent if it has a ?nite limit i.e., lim
n?8
a
n
= L= ?nite
unique limit value.
Divergent If lim
n?8
a
n
= in?nite=±8.
Oscillatory If limit ofa
n
is not unique (oscillates
?nitely) or±8 (oscillates in?nitely).
Examples:
1.

1
n
2

convergent since lim
n?8
1
n
2
= 0=?nite unique
2. {n:} divergent since lim
n?8
n :=8
3. {(-1)
n
} oscillates ?nitely, since
lim
n?8
(-1)
n
=

1,n even
-1,n odd.
4. {(-1)
n
·n
2
} oscillates in?nitely,
since limit=±8.
Result1: If sequence{a
n
} converges to limitL and
{b
n
} converges toL
*
then
a. {a
n
+b
n
} converges toL+L
*
b. {ca
n
} converges toCL
c. {a
n
·b
n
} converges toL·L
*
d. {
a
n
b
n
} converges to
L
L
*
, providedL
*
= 0.
Result2: Every convergent sequence is bounded.
Example:

1
n

is convergent and is bounded
a
n
=
1
n
< 1,forevery n.
Result3: The converse is not true i.e., a bounded
sequence may not be convergent.
Example:{(-1)
n
} is oscillatory (has more than one
limit but is bounded since-1= (-1)
n
= 1.
Result4: A bounded monotonic sequence is con-
vergent.
Example:

1
n
2

is bounded since
1
n
2
=1forevery n
and monotonically decreasing since
1
n
2
>
1
(n+1)
2
for
every n. Hence the sequence is convergent because
lim
n?8
a
n
= lim
n?8
1
n
2
= 0= ?nite.
Useful Standard Limits
1. a. lim
n?8
1
n
= 0, b. lim
n?8
1
n
2
= 0, c. lim
n?8
1
v
n
= 0
2. lim
n?8
n
1/n
= 1
3. lim
n?8
logn
n
= 0
4. lim
n?8

1+
x
n

n
= e
x
, forany x
5. lim
n?8
x
1/n
=1forx> 0
6. (a) lim
n?8
x
n
=0for |x| <1i.e. - 1<x<1.
(b) lim
n?8
x
n
n!
=0 forany x. In formulas (5) and
6(b)x remains ?xed asn?8
WORKED OUT EXAMPLES
Determine the nature of the following sequences
whose nth term a
n
is
chap-02 B.V.Ramana August 30, 2006 10:15
SEQUENCESANDSERIES 2.3
Example1: a
n
=
n
2
-n
2n
2
+n
Solution:
lim
n?8
a
n
= lim
n?8
n
2
-n
2n
2
+n
= lim
n?8
1-
1
n
2+
1
n
=
1
2
sequence is convergent since the limit of the se-
quence is unique and ?nite.
Example2: a
n
= tanh n.
Solution:
lim
n?8
a
n
= lim
n?8
tanh n= lim
n?8
sinh n
cosh n
= lim
n?8
e
n
-e
-n
e
n
+e
-n
= lim
n?8
e
2n
- 1
e
2n
+ 1
= lim
n?8
1-
1
e
2n
1+
1
e
2n
= 1 so convergent.
Example3: a
n
= e
n
.
Solution: lim
n?8
e
n
=8 so divergent.
Example4: a
n
= 2+ (-1)
n
.
Solution:
lim
n?8
a
2n
= lim
n?8
{2+ (-1)
2n
}= 2+ 1= 3
lim
n?8
a
2n-1
= lim
n?8
{2+ (-1)
2n-1
}= 2- 1= 1
sequence oscillates ?nitely since it has more than
one ?nite (two) limits.
EXERCISE
1.
2n+1
1-3n
Ans. convergent, limit=
-2
3
2. 1+
(-1)
n
n
Ans. convergent, limit= 1
3.
1+(-1)
n
n
Ans. convergent, limit= 0
4. sinn Ans. divergent, limit=8
5.
lnn
n
Ans. convergent, limit= 0
Hint: Apply L’ Hospital’s rule.
6.
1
3
n
Ans. convergent, limit=
3
2
7.
(-1)
n-1
n
3
n
Ans. convergent
8.

n
n+1
	
2
Ans. convergent
9.
(n+1)
2
(n+1)!
Ans. convergent
10. 2n Ans. divergent, limit=8
11. 1+
1
n
Ans. convergent, limit= 1
12. [n+ (-1)
n
]
-1
Ans. convergent.
2.4 INFINITE SERIES
Differential Equations are frequently solved by
using in?nite series. Fourier series, Fourier-Bessel
series, etc. expansions involve in?nite series. Tran-
scendental functions (trigonometric, exponential,
logarithmic, hyperpolic, etc.) can be expressed
conveniently in terms of in?nite series. Many prob-
lems that cannot be solved in terms of elementary
(algebraic and transcendental) functions can also be
solved in terms of in?nite series.
Series
Given a sequence of numbers u
1
,u
2
,u
3
,...u
n
,...
the expression
u
1
+u
2
+u
3
+···+u
n
+··· (1)
which is the sum of the terms of the sequence, is
known as a numerical series or simply “series”. The
numbers u
1
,u
2
,u
3
,...u
n
are known as the ?rst,
second, third,...,nth term of the series (1).
In?nite Series
Ifthenumberoftermsintheseries(1)isin?nite,then
the series is called an in?nite series (otherwise ?nite
series when the number of terms is ?nite). In?nite
series (1) is usually denoted as
8


n=1
u
n
or


u
n
(1)
Themainaimofthischapteristostudythenature(or
behaviour) of convergence, divergence or oscillation
ofagivenin?niteseries.Forthispurpose,de?ne{S
n
}
Page 4


chap-02 B.V.Ramana August 30, 2006 10:15
Chapter2
Sequences and Series
INTRODUCTION
The study of convergence and divergence of a se-
quence, which is an ordered list of things, is a prereq-
uisit for in?nite series. The unit square in the ?gure
can be expressed as an in?nite (geometric) series
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+···
Several functions can be expressed as “in?nite
polynomials” (known as “powerser ies”) using
the concept of in?nite series. By Fourier series,
certain functions can be represented as an in?nite
sum of trigonometric functions. Using in?nite
series, differential equations in problems of signal
transmission, chemical diffusion, vibration and heat
?ow can be solved and non elementary integrals
evaluated. The in?nite process of summing of an
in?nite series is a puzzle for centuries convergence
and divergence of in?nite series plays an important
role in engineering applications.
1
2
1
4
1
8
1
16
2.1 SEQUENCES
A sequence is a function from the domain set of
natural numbersN to any setS.
Real sequence isafunctionfromN to R,
the set of real numbers; denoted by f :N ? R.
Thus the real sequence f is set of all ordered
pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs
(n,f (n)) with n a positive integer.
Notation: Sincethedomainofasequenceisalways
thesame(thesetofpositiveintegers)asequencemay
be written as{f (n)} instead of{n,f (n)}.
Examples:
1.

n,
1
n

=

1
n

=

1,
1
2
,
1
3
,
1
4
,...
1
n
...

2.

n,
1
2
n-1

=

1
2
n-1

=

1,
1
2
,
1
2
2
,
1
2
3
,
1
2
4
,...,
1
2
n-1
,...

constant sequence where range is singleton set{c},
c= constant.
Example: {3, 3, 3, 3,...}
Null sequence{0, 0, 0,..., 0,...}
A sequence is also denoted by{a
n
} whose ordinate
y = a
n
at the abscissax = n. Thus in a sequence for
each positive integern, a numbera
n
is assigned and
is denoted as	a
n

or( a
n
)or
{a
n
}={a(1),a(2),a(3),...,a(n),...}
={a
1
,a
2
,a
3
,...,a
n
,...}
Herea
1
,a
2
,a
3
,...a
n
, are known as the ?rst, second,
third and nth terms of the sequence.
In?nite sequence is a sequence in which the
numberoftermsisin?nite,andisdenotedby {a
n
}
8
n=1
.
Ontheotherhand,?nitesequencedenotedby{a
n
}
m
n=1
containsonlya?nitenumberofterms( m=?nite).
Bounded sequence A sequence {a
n
} is said to
be bounded if there exists numbers m and M such
thatm<a
n
<M forevery n, otherwise it is said to
be unbounded.
Monotonic sequence
A sequence{a
n
}issaidtobe
a. monotonically increasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
2.1
chap-02 B.V.Ramana August 30, 2006 10:15
2.2 ENGINEERING MATHEMATICS
b. monotonically decreasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
c.monotonic if it is eithermonotonically
increasing or monotonically decreasing.
Example:

1
n

=

1,
1
2
,
1
3
,
1
4
,
1
5
,...

bounded
since 0<a
n
=
1
n
< 1 and monotonically decreasing.
Example: {2
n
}={2, 2
2
, 2
3
, 2
4
,...} unbounded
since 2
n
becomes larger and larger asn comes large
and monotonically increasing.
2.2 LIMIT OF A SEQUENCE
Considerasequence {a
n
}=

3+
1
n

.
Plotting the values
n: 1 2 4 5 10 50 100 1000 10000 100000...
a
n
: 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001...
As n increases, a
n
= 3+
1
n
becomescloserto3.
Thus the difference (or distance) between 3+
1
n
and
3becomessmallerandsmalleras n becomes larger
and larger i.e., we can make 3+
1
3
and3asclose
as we please, by choosing an appropriately (suf?-
ciently) large value forn, i.e., the terms of a sequence
clusteraroundthis(limit)point.Howevernotethat
3+
1
n
=3foranyvalueof n.
Limit: A number L is said to be a limit of a
sequence{a
n
} and is denoted as
lim
As n?8
a
n
= lim
n?8
a
n
= lim a
n
= L
ifforevery > 0 there existsN such that
|a
n
-L|< forall n=N.
Note: Asequencemayhaveauniquelimitormay
havemorethanonelimitormaynothavealimitatall.
Result: A monotonic sequence always has a limit
(maybe?niteorin?nite).
2.3 CONVERGENCE, DIVERGENCE AND
OSCILLATION OF A SEQUENCE
Convergent A sequence{a
n
} is said to be conver-
gent if it has a ?nite limit i.e., lim
n?8
a
n
= L= ?nite
unique limit value.
Divergent If lim
n?8
a
n
= in?nite=±8.
Oscillatory If limit ofa
n
is not unique (oscillates
?nitely) or±8 (oscillates in?nitely).
Examples:
1.

1
n
2

convergent since lim
n?8
1
n
2
= 0=?nite unique
2. {n:} divergent since lim
n?8
n :=8
3. {(-1)
n
} oscillates ?nitely, since
lim
n?8
(-1)
n
=

1,n even
-1,n odd.
4. {(-1)
n
·n
2
} oscillates in?nitely,
since limit=±8.
Result1: If sequence{a
n
} converges to limitL and
{b
n
} converges toL
*
then
a. {a
n
+b
n
} converges toL+L
*
b. {ca
n
} converges toCL
c. {a
n
·b
n
} converges toL·L
*
d. {
a
n
b
n
} converges to
L
L
*
, providedL
*
= 0.
Result2: Every convergent sequence is bounded.
Example:

1
n

is convergent and is bounded
a
n
=
1
n
< 1,forevery n.
Result3: The converse is not true i.e., a bounded
sequence may not be convergent.
Example:{(-1)
n
} is oscillatory (has more than one
limit but is bounded since-1= (-1)
n
= 1.
Result4: A bounded monotonic sequence is con-
vergent.
Example:

1
n
2

is bounded since
1
n
2
=1forevery n
and monotonically decreasing since
1
n
2
>
1
(n+1)
2
for
every n. Hence the sequence is convergent because
lim
n?8
a
n
= lim
n?8
1
n
2
= 0= ?nite.
Useful Standard Limits
1. a. lim
n?8
1
n
= 0, b. lim
n?8
1
n
2
= 0, c. lim
n?8
1
v
n
= 0
2. lim
n?8
n
1/n
= 1
3. lim
n?8
logn
n
= 0
4. lim
n?8

1+
x
n

n
= e
x
, forany x
5. lim
n?8
x
1/n
=1forx> 0
6. (a) lim
n?8
x
n
=0for |x| <1i.e. - 1<x<1.
(b) lim
n?8
x
n
n!
=0 forany x. In formulas (5) and
6(b)x remains ?xed asn?8
WORKED OUT EXAMPLES
Determine the nature of the following sequences
whose nth term a
n
is
chap-02 B.V.Ramana August 30, 2006 10:15
SEQUENCESANDSERIES 2.3
Example1: a
n
=
n
2
-n
2n
2
+n
Solution:
lim
n?8
a
n
= lim
n?8
n
2
-n
2n
2
+n
= lim
n?8
1-
1
n
2+
1
n
=
1
2
sequence is convergent since the limit of the se-
quence is unique and ?nite.
Example2: a
n
= tanh n.
Solution:
lim
n?8
a
n
= lim
n?8
tanh n= lim
n?8
sinh n
cosh n
= lim
n?8
e
n
-e
-n
e
n
+e
-n
= lim
n?8
e
2n
- 1
e
2n
+ 1
= lim
n?8
1-
1
e
2n
1+
1
e
2n
= 1 so convergent.
Example3: a
n
= e
n
.
Solution: lim
n?8
e
n
=8 so divergent.
Example4: a
n
= 2+ (-1)
n
.
Solution:
lim
n?8
a
2n
= lim
n?8
{2+ (-1)
2n
}= 2+ 1= 3
lim
n?8
a
2n-1
= lim
n?8
{2+ (-1)
2n-1
}= 2- 1= 1
sequence oscillates ?nitely since it has more than
one ?nite (two) limits.
EXERCISE
1.
2n+1
1-3n
Ans. convergent, limit=
-2
3
2. 1+
(-1)
n
n
Ans. convergent, limit= 1
3.
1+(-1)
n
n
Ans. convergent, limit= 0
4. sinn Ans. divergent, limit=8
5.
lnn
n
Ans. convergent, limit= 0
Hint: Apply L’ Hospital’s rule.
6.
1
3
n
Ans. convergent, limit=
3
2
7.
(-1)
n-1
n
3
n
Ans. convergent
8.

n
n+1
	
2
Ans. convergent
9.
(n+1)
2
(n+1)!
Ans. convergent
10. 2n Ans. divergent, limit=8
11. 1+
1
n
Ans. convergent, limit= 1
12. [n+ (-1)
n
]
-1
Ans. convergent.
2.4 INFINITE SERIES
Differential Equations are frequently solved by
using in?nite series. Fourier series, Fourier-Bessel
series, etc. expansions involve in?nite series. Tran-
scendental functions (trigonometric, exponential,
logarithmic, hyperpolic, etc.) can be expressed
conveniently in terms of in?nite series. Many prob-
lems that cannot be solved in terms of elementary
(algebraic and transcendental) functions can also be
solved in terms of in?nite series.
Series
Given a sequence of numbers u
1
,u
2
,u
3
,...u
n
,...
the expression
u
1
+u
2
+u
3
+···+u
n
+··· (1)
which is the sum of the terms of the sequence, is
known as a numerical series or simply “series”. The
numbers u
1
,u
2
,u
3
,...u
n
are known as the ?rst,
second, third,...,nth term of the series (1).
In?nite Series
Ifthenumberoftermsintheseries(1)isin?nite,then
the series is called an in?nite series (otherwise ?nite
series when the number of terms is ?nite). In?nite
series (1) is usually denoted as
8


n=1
u
n
or


u
n
(1)
Themainaimofthischapteristostudythenature(or
behaviour) of convergence, divergence or oscillation
ofagivenin?niteseries.Forthispurpose,de?ne{S
n
}
chap-02 B.V.Ramana August 30, 2006 10:15
2.4 ENGINEERING MATHEMATICS
the sequence of partial sums as
S
1
=u
1
S
2
=u
1
+u
2
S
3
=u
1
+u
2
+u
3
.
.
.
S
n
=u
1
+u
2
+u
3
+···+u
n
=
n


k=1
u
k
Here S
n
is known as the nth partial sum of the
series, i.e., it is the sum of the ?rst n terms of the
series (1).
Convergence
An in?nite series
8

n=1
u
n
is said to be convergent if
8


n=1
u
n
= lim
n?8

n


k=1
u
k

= lim
n?8
S
n
=?nite limit value=S
HereS is known as the sum (value) of the series (1).
Divergence
If lim
n?8
S
n
does not exist (i.e., lim
n?8
S
n
=±8) then
series (1) is said to be divergent.
Oscillation
When lim
n?8
S
n
tends to more than one limit (non
unique) orto ±8 then series (1) is said to be os-
cillatory. Thus the behaviour of convergence, diver-
genceoroscillationofaseriesisthebahaviourofits
sequence of partial sums{S
n
}.
Example: 1+
1
4
+
1
16
+
1
64
+···
Hereu
n
=
1
4
n-1
so lim
n?8
S
n
= lim
n?8
1-
1
4
n
1-
1
4
=
lim
n?8
4
3

1-
1
4
n
	
=
4
3
= ?nite,
series converges.
Example: 1
2
+ 2
2
+ 3
2
+···+n
2
+···
lim
n?8
S
n
= lim
n?8
n(n+1)(2n+1)
6
=8, series diverges.
Example: 7- 4- 3+ 7- 4- 3+ 7- 4- 3+···
lim
n?8
S
n
=0 or7 or3 accor ding as the numberof
terms is 3m, 3m+1or3m+ 2.
Since the limit is not unique, series oscillates
(?nitely).
Example: 1-2+3-4+···+(-1)
n-1
n+···
lim
n?8
S
n
=-
n
2
=-8 if n is even
lim
n?8
S
n
=
n+ 1
2
=+8 if n is odd
series oscillates (in?nitely).
Some General Properties of Series
1. If a series

u
n
converges to a sum s then the
seriesc

u
n
also converges to the sumcs, where
c is a constant.
2. Iftheseries

u
n
and

v
n
converges to the sums
s

ands

respectively then the series

(u
n
+v
n
)
and

(u
n
-v
n
) also converge to s

+s

and
s

–s

respectively. Addition or subtraction of two
series is done by termwise addition or termwise
subtraction.
3. The convergence of a series is not affected by
the suppression (deletion) or addition of a ?nite
numberofitsterms,sincethedeletionoraddition
ofthesumofthese?nitenumberofterms(which
isa?nitequantity)doesnotalterthebehaviour
of the sum of the series.
2.5 NECESSARY CONDITION
FOR CONVERGENCE
Necessary condition for convergence of a series

u
n
is that, its nth term u
n
approaches zero as n
becomes in?nite i.e.,
If series converges, then lim
n?8
u
n
= 0.
Important Note: Thisisnotatestforconvergence.
Proof: Let s be the sum of this convergent series.
Also letS
n
andS
n-1
be the nth and (n- 1)th partial
sums of the given series so that
u
n
= S
n
-S
n-1
Taking limit, we have
lim
n?8
u
n
= lim
n?8
(S
n
-S
n-1
)= lim
n?8
S
n
- lim
n?8
S
n-1
=s-s = 0.
Note 1: The converse of the above result is not
true, i.e., the above result is not a suf?cient condition.
From the fact that the nth term u
n
approaches zero,
Page 5


chap-02 B.V.Ramana August 30, 2006 10:15
Chapter2
Sequences and Series
INTRODUCTION
The study of convergence and divergence of a se-
quence, which is an ordered list of things, is a prereq-
uisit for in?nite series. The unit square in the ?gure
can be expressed as an in?nite (geometric) series
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+···
Several functions can be expressed as “in?nite
polynomials” (known as “powerser ies”) using
the concept of in?nite series. By Fourier series,
certain functions can be represented as an in?nite
sum of trigonometric functions. Using in?nite
series, differential equations in problems of signal
transmission, chemical diffusion, vibration and heat
?ow can be solved and non elementary integrals
evaluated. The in?nite process of summing of an
in?nite series is a puzzle for centuries convergence
and divergence of in?nite series plays an important
role in engineering applications.
1
2
1
4
1
8
1
16
2.1 SEQUENCES
A sequence is a function from the domain set of
natural numbersN to any setS.
Real sequence isafunctionfromN to R,
the set of real numbers; denoted by f :N ? R.
Thus the real sequence f is set of all ordered
pairs{n,f (n)}|{n= 1, 2, 3,...} i.e., set of all pairs
(n,f (n)) with n a positive integer.
Notation: Sincethedomainofasequenceisalways
thesame(thesetofpositiveintegers)asequencemay
be written as{f (n)} instead of{n,f (n)}.
Examples:
1.

n,
1
n

=

1
n

=

1,
1
2
,
1
3
,
1
4
,...
1
n
...

2.

n,
1
2
n-1

=

1
2
n-1

=

1,
1
2
,
1
2
2
,
1
2
3
,
1
2
4
,...,
1
2
n-1
,...

constant sequence where range is singleton set{c},
c= constant.
Example: {3, 3, 3, 3,...}
Null sequence{0, 0, 0,..., 0,...}
A sequence is also denoted by{a
n
} whose ordinate
y = a
n
at the abscissax = n. Thus in a sequence for
each positive integern, a numbera
n
is assigned and
is denoted as	a
n

or( a
n
)or
{a
n
}={a(1),a(2),a(3),...,a(n),...}
={a
1
,a
2
,a
3
,...,a
n
,...}
Herea
1
,a
2
,a
3
,...a
n
, are known as the ?rst, second,
third and nth terms of the sequence.
In?nite sequence is a sequence in which the
numberoftermsisin?nite,andisdenotedby {a
n
}
8
n=1
.
Ontheotherhand,?nitesequencedenotedby{a
n
}
m
n=1
containsonlya?nitenumberofterms( m=?nite).
Bounded sequence A sequence {a
n
} is said to
be bounded if there exists numbers m and M such
thatm<a
n
<M forevery n, otherwise it is said to
be unbounded.
Monotonic sequence
A sequence{a
n
}issaidtobe
a. monotonically increasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
2.1
chap-02 B.V.Ramana August 30, 2006 10:15
2.2 ENGINEERING MATHEMATICS
b. monotonically decreasing if a
n+1
= a
n
for
every n
i.e., a
1
= a
2
= a
3
=···= a
n
= a
n+1
=···
c.monotonic if it is eithermonotonically
increasing or monotonically decreasing.
Example:

1
n

=

1,
1
2
,
1
3
,
1
4
,
1
5
,...

bounded
since 0<a
n
=
1
n
< 1 and monotonically decreasing.
Example: {2
n
}={2, 2
2
, 2
3
, 2
4
,...} unbounded
since 2
n
becomes larger and larger asn comes large
and monotonically increasing.
2.2 LIMIT OF A SEQUENCE
Considerasequence {a
n
}=

3+
1
n

.
Plotting the values
n: 1 2 4 5 10 50 100 1000 10000 100000...
a
n
: 4 3.5 3.25 3.2 3.1 3.02 3.01 3.001 3.0001 3.00001...
As n increases, a
n
= 3+
1
n
becomescloserto3.
Thus the difference (or distance) between 3+
1
n
and
3becomessmallerandsmalleras n becomes larger
and larger i.e., we can make 3+
1
3
and3asclose
as we please, by choosing an appropriately (suf?-
ciently) large value forn, i.e., the terms of a sequence
clusteraroundthis(limit)point.Howevernotethat
3+
1
n
=3foranyvalueof n.
Limit: A number L is said to be a limit of a
sequence{a
n
} and is denoted as
lim
As n?8
a
n
= lim
n?8
a
n
= lim a
n
= L
ifforevery > 0 there existsN such that
|a
n
-L|< forall n=N.
Note: Asequencemayhaveauniquelimitormay
havemorethanonelimitormaynothavealimitatall.
Result: A monotonic sequence always has a limit
(maybe?niteorin?nite).
2.3 CONVERGENCE, DIVERGENCE AND
OSCILLATION OF A SEQUENCE
Convergent A sequence{a
n
} is said to be conver-
gent if it has a ?nite limit i.e., lim
n?8
a
n
= L= ?nite
unique limit value.
Divergent If lim
n?8
a
n
= in?nite=±8.
Oscillatory If limit ofa
n
is not unique (oscillates
?nitely) or±8 (oscillates in?nitely).
Examples:
1.

1
n
2

convergent since lim
n?8
1
n
2
= 0=?nite unique
2. {n:} divergent since lim
n?8
n :=8
3. {(-1)
n
} oscillates ?nitely, since
lim
n?8
(-1)
n
=

1,n even
-1,n odd.
4. {(-1)
n
·n
2
} oscillates in?nitely,
since limit=±8.
Result1: If sequence{a
n
} converges to limitL and
{b
n
} converges toL
*
then
a. {a
n
+b
n
} converges toL+L
*
b. {ca
n
} converges toCL
c. {a
n
·b
n
} converges toL·L
*
d. {
a
n
b
n
} converges to
L
L
*
, providedL
*
= 0.
Result2: Every convergent sequence is bounded.
Example:

1
n

is convergent and is bounded
a
n
=
1
n
< 1,forevery n.
Result3: The converse is not true i.e., a bounded
sequence may not be convergent.
Example:{(-1)
n
} is oscillatory (has more than one
limit but is bounded since-1= (-1)
n
= 1.
Result4: A bounded monotonic sequence is con-
vergent.
Example:

1
n
2

is bounded since
1
n
2
=1forevery n
and monotonically decreasing since
1
n
2
>
1
(n+1)
2
for
every n. Hence the sequence is convergent because
lim
n?8
a
n
= lim
n?8
1
n
2
= 0= ?nite.
Useful Standard Limits
1. a. lim
n?8
1
n
= 0, b. lim
n?8
1
n
2
= 0, c. lim
n?8
1
v
n
= 0
2. lim
n?8
n
1/n
= 1
3. lim
n?8
logn
n
= 0
4. lim
n?8

1+
x
n

n
= e
x
, forany x
5. lim
n?8
x
1/n
=1forx> 0
6. (a) lim
n?8
x
n
=0for |x| <1i.e. - 1<x<1.
(b) lim
n?8
x
n
n!
=0 forany x. In formulas (5) and
6(b)x remains ?xed asn?8
WORKED OUT EXAMPLES
Determine the nature of the following sequences
whose nth term a
n
is
chap-02 B.V.Ramana August 30, 2006 10:15
SEQUENCESANDSERIES 2.3
Example1: a
n
=
n
2
-n
2n
2
+n
Solution:
lim
n?8
a
n
= lim
n?8
n
2
-n
2n
2
+n
= lim
n?8
1-
1
n
2+
1
n
=
1
2
sequence is convergent since the limit of the se-
quence is unique and ?nite.
Example2: a
n
= tanh n.
Solution:
lim
n?8
a
n
= lim
n?8
tanh n= lim
n?8
sinh n
cosh n
= lim
n?8
e
n
-e
-n
e
n
+e
-n
= lim
n?8
e
2n
- 1
e
2n
+ 1
= lim
n?8
1-
1
e
2n
1+
1
e
2n
= 1 so convergent.
Example3: a
n
= e
n
.
Solution: lim
n?8
e
n
=8 so divergent.
Example4: a
n
= 2+ (-1)
n
.
Solution:
lim
n?8
a
2n
= lim
n?8
{2+ (-1)
2n
}= 2+ 1= 3
lim
n?8
a
2n-1
= lim
n?8
{2+ (-1)
2n-1
}= 2- 1= 1
sequence oscillates ?nitely since it has more than
one ?nite (two) limits.
EXERCISE
1.
2n+1
1-3n
Ans. convergent, limit=
-2
3
2. 1+
(-1)
n
n
Ans. convergent, limit= 1
3.
1+(-1)
n
n
Ans. convergent, limit= 0
4. sinn Ans. divergent, limit=8
5.
lnn
n
Ans. convergent, limit= 0
Hint: Apply L’ Hospital’s rule.
6.
1
3
n
Ans. convergent, limit=
3
2
7.
(-1)
n-1
n
3
n
Ans. convergent
8.

n
n+1
	
2
Ans. convergent
9.
(n+1)
2
(n+1)!
Ans. convergent
10. 2n Ans. divergent, limit=8
11. 1+
1
n
Ans. convergent, limit= 1
12. [n+ (-1)
n
]
-1
Ans. convergent.
2.4 INFINITE SERIES
Differential Equations are frequently solved by
using in?nite series. Fourier series, Fourier-Bessel
series, etc. expansions involve in?nite series. Tran-
scendental functions (trigonometric, exponential,
logarithmic, hyperpolic, etc.) can be expressed
conveniently in terms of in?nite series. Many prob-
lems that cannot be solved in terms of elementary
(algebraic and transcendental) functions can also be
solved in terms of in?nite series.
Series
Given a sequence of numbers u
1
,u
2
,u
3
,...u
n
,...
the expression
u
1
+u
2
+u
3
+···+u
n
+··· (1)
which is the sum of the terms of the sequence, is
known as a numerical series or simply “series”. The
numbers u
1
,u
2
,u
3
,...u
n
are known as the ?rst,
second, third,...,nth term of the series (1).
In?nite Series
Ifthenumberoftermsintheseries(1)isin?nite,then
the series is called an in?nite series (otherwise ?nite
series when the number of terms is ?nite). In?nite
series (1) is usually denoted as
8


n=1
u
n
or


u
n
(1)
Themainaimofthischapteristostudythenature(or
behaviour) of convergence, divergence or oscillation
ofagivenin?niteseries.Forthispurpose,de?ne{S
n
}
chap-02 B.V.Ramana August 30, 2006 10:15
2.4 ENGINEERING MATHEMATICS
the sequence of partial sums as
S
1
=u
1
S
2
=u
1
+u
2
S
3
=u
1
+u
2
+u
3
.
.
.
S
n
=u
1
+u
2
+u
3
+···+u
n
=
n


k=1
u
k
Here S
n
is known as the nth partial sum of the
series, i.e., it is the sum of the ?rst n terms of the
series (1).
Convergence
An in?nite series
8

n=1
u
n
is said to be convergent if
8


n=1
u
n
= lim
n?8

n


k=1
u
k

= lim
n?8
S
n
=?nite limit value=S
HereS is known as the sum (value) of the series (1).
Divergence
If lim
n?8
S
n
does not exist (i.e., lim
n?8
S
n
=±8) then
series (1) is said to be divergent.
Oscillation
When lim
n?8
S
n
tends to more than one limit (non
unique) orto ±8 then series (1) is said to be os-
cillatory. Thus the behaviour of convergence, diver-
genceoroscillationofaseriesisthebahaviourofits
sequence of partial sums{S
n
}.
Example: 1+
1
4
+
1
16
+
1
64
+···
Hereu
n
=
1
4
n-1
so lim
n?8
S
n
= lim
n?8
1-
1
4
n
1-
1
4
=
lim
n?8
4
3

1-
1
4
n
	
=
4
3
= ?nite,
series converges.
Example: 1
2
+ 2
2
+ 3
2
+···+n
2
+···
lim
n?8
S
n
= lim
n?8
n(n+1)(2n+1)
6
=8, series diverges.
Example: 7- 4- 3+ 7- 4- 3+ 7- 4- 3+···
lim
n?8
S
n
=0 or7 or3 accor ding as the numberof
terms is 3m, 3m+1or3m+ 2.
Since the limit is not unique, series oscillates
(?nitely).
Example: 1-2+3-4+···+(-1)
n-1
n+···
lim
n?8
S
n
=-
n
2
=-8 if n is even
lim
n?8
S
n
=
n+ 1
2
=+8 if n is odd
series oscillates (in?nitely).
Some General Properties of Series
1. If a series

u
n
converges to a sum s then the
seriesc

u
n
also converges to the sumcs, where
c is a constant.
2. Iftheseries

u
n
and

v
n
converges to the sums
s

ands

respectively then the series

(u
n
+v
n
)
and

(u
n
-v
n
) also converge to s

+s

and
s

–s

respectively. Addition or subtraction of two
series is done by termwise addition or termwise
subtraction.
3. The convergence of a series is not affected by
the suppression (deletion) or addition of a ?nite
numberofitsterms,sincethedeletionoraddition
ofthesumofthese?nitenumberofterms(which
isa?nitequantity)doesnotalterthebehaviour
of the sum of the series.
2.5 NECESSARY CONDITION
FOR CONVERGENCE
Necessary condition for convergence of a series

u
n
is that, its nth term u
n
approaches zero as n
becomes in?nite i.e.,
If series converges, then lim
n?8
u
n
= 0.
Important Note: Thisisnotatestforconvergence.
Proof: Let s be the sum of this convergent series.
Also letS
n
andS
n-1
be the nth and (n- 1)th partial
sums of the given series so that
u
n
= S
n
-S
n-1
Taking limit, we have
lim
n?8
u
n
= lim
n?8
(S
n
-S
n-1
)= lim
n?8
S
n
- lim
n?8
S
n-1
=s-s = 0.
Note 1: The converse of the above result is not
true, i.e., the above result is not a suf?cient condition.
From the fact that the nth term u
n
approaches zero,
chap-02 B.V.Ramana August 30, 2006 10:15
SEQUENCESANDSERIES 2.5
it does not follow that the series converges, for the
series may diverge.
If lim
n?8
u
n
= 0, then the series may converge or
may diverge.
Example: 1+
1
2
+
1
3
+
1
4
+···+
1
n
+···isadi-
vergent series although itsnth term approaches zero
i.e.,
lim
n?8
u
n
= lim
n?8
1
n
= 0
Note 2: Preliminary test for divergence.
If thenth term of a series does not tend to zero as
n?8, then the series diverges i.e.,
if lim
n?8
u
n
= 0 then series diverges.
Example:
1
2
+
2
3
+
3
4
+
4
5
+···+
n
n+ 1
+···
Since lim
n?8
u
n
= lim
n?8
n
n+1
= 1= 0 by the above
preliminary test, the given series diverges.
2.6 STANDARD INFINITE SERIES:
GEOMETRIC SERIES AND
HARMONIC SERIES
Geometric Series Test
8


n=0
ar
n
= a+ar+ar
2
+ar
3
+···+ar
n-1
+···,
with a =0(1)
is a geometric series, whose terms form a geometric
progression with the ?rst term a and the common
ratio r.Forthisseries
S
n
=
a-ar
n
1-r
=
a
1-r
-
ar
n
1-r
Case1: When|r| < 1 then lim
n?8
r
n
= 0 so that
lim
n?8
S
n
= lim
n?8

a
1-r
-
ar
n
1-r

=
a
1-r
-
a
1-r
· 0
=
a
1-r
= ?nite
Hence geometric series (1) converges to the
sum
a
1-r
when |r| < 1 i.e., in the interval -1 <
r< 1.
Case2: When|r| > 1 then lim
n?8
r
n
=8 so that
lim
n?8
S
n
= lim
n?8

a
1-r
-
ar
n
1-r

=±8
Thus series (1) diverges when|r| > 1
i.e., whenr>1orr<-1.
Case3: If r = 1, the series (1) reduces to
a+a+a+···
consequently lim
n?8
S
n
= lim
n?8
(na)=8
so series diverges.
Case4: If r=-1, the series (1) reduces to
a-a+a-a+···
In this case,
Sn=

0, when n is even
a, when n is odd
Thus lim
n?8
S
n
is not unique (more than one limit)
hence the series diverges.
Hence the geometric series converges only when
|r| <1anddivergesforallothervaluesof r.
Example: A ball is dropped from a height b feet
from a ?at surface. Each time the ball hits the ground
afterfalling a distance h it rebounds a distance rh
where 0<r< 1 (Fig. 2.1).
Fig. 2.1
Find the total distance the ball travels if b=4ft
andr =
3
4
.
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