Chapter 1 Sequences and Series - BS Publications
Chapter 1
Sequences and Series
? Sequences and Series
? Convergence of Infinite Series
? Tests of Convergence
? P-Series Test ? Comparison Tests ? Ratio test ? Raabe's test ? Cauchy's Root test ? Integral test ? Leibnitz's test
? Absolute Convergence
? Conditional convergence ? Power series and Interval of convergence
? Summary of all Tests
? Solved University Questions (JNTU)
? Objective type of Questions
2
Engineering Mathematics - I
1.1 Sequence
A function f:N S, where S is any nonempty set is called a Sequence i.e., for each n N, a unique element f(n) S. The sequence is written as f(1), f(2), f(3), ......f(n)...., and is denoted by {f(n)}, or , or (f(n)). If f(n) = an , the sequence is
written as a1, a2.....an and denoted by , {an}or < an > or (an ). Here f(n) or an are the
nth terms of the Sequence.
Ex. 1. 1 , 4 , 9 , 16 ,......... n2 ,.....(or) < n2 >
N
S
1
2
4
3
. .
.9 .
n
n2
.
.
.
.
Ex. 2.
1 13
,
1 23
,
1 33
,.....
1 n3
....(or
)
1 n 3
Ex. 3.
1, 1, 1......1..... or
Sequences and Series
3
Ex 4: 1 , ?1, 1, ?1, ......... or (-1)n-1
Note : 1. If S R then the sequence is called a real sequence.
2. The range of a sequence is almost a countable set.
1.1.1 Kinds of Sequences
1. Finite Sequence: A sequence < an > in which an = 0 n > m N is said to
be a finite Sequence. i.e., A finite Sequence has a finite number of terms. 2. Infinite Sequence: A sequence, which is not finite, is an infinite sequence.
1.1.2 Bounds of a Sequence and Bounded Sequence
1. If a number `M' an M, n N, the Sequence < an > is said to be
bounded above or bounded on the right.
Ex.
1, 1 , 1 , 23
,.......
here
an
1
n N
2. If a number `m' an m, n N, the sequence < an > is said to be
bounded below or bounded on the left.
Ex. 1 , 2 , 3 ,.....here an 1 n N
3. A sequence which is bounded above and below is said to be bounded.
Ex.
Let
an
= (-1)n 1+
1 n
n
1
2
3
4 ......
a n
-2 3/2 -4/3 5/4 ......
4
Engineering Mathematics - I
From the above figure (see also table) it can be seen that m = ?2 and M = 3 . 2
The sequence is bounded.
1.1.3 Limits of a Sequence
A Sequence < an > is said to tend to limit `l' when, given any + ve number ' ',
however small, we can always find an integer `m' such that an - l 1 . 2
1.1.4 Convergent, Divergent and Oscillatory Sequences
1. Convergent Sequence: A sequence which tends to a finite limit, say `l' is called a Convergent Sequence. We say that the sequence converges to `l'
2. Divergent Sequence: A sequence which tends to ? is said to be Divergent
(or is said to diverge).
3. Oscillatory Sequence: A sequence which neither converges nor diverges ,is
called an Oscillatory Sequence.
Ex. 1.
Consider the sequence
2 ,
3 , 4 , 5 ,..... here 234
an
=1+
1 n
The sequence < an > is convergent and has the limit 1
an
-1=1+
1 n
-1
=
1 n
and
1 < whenever n
n>1
Suppose we choose = .001 , we have 1 < .001 when n > 1000. n
Ex. 2.
If
an
= 3 + (-1)n
1 'n
'
<
an
>
converges to 3.
Sequences and Series
5
Ex. 3. Ex. 4.
If an = n2 + (-1)n .n,< an > diverges.
If
an
=
1 n
+
2 ( -1)n
,<
an
>
oscillates between -2 and 2.
1.2 Infinite Series
If < un > is a sequence, then the expression u1 + u2 + u3 + ........ + un + ..... is called an
infinite series. It is denoted by
n=1
u n
or simply
u n
The sum of the first n terms of the series is denoted by sn
i.e., sn = u1 + u2 + u3 + ...... + un ; s1, s2 , s3,....sn are called partial sums.
1.2.1 Convergent, Divergent and Oscillatory Series
Let un be an infinite series. As n , there are three possibilities. (a) Convergent series: As n , sn a finite limit, say `s' in which case the series is said to be convergent and `s' is called its sum to infinity.
Thus
Lt
n
sn
=s
(or) simply
Ltsn
= s
This
is
also
written
as
u1 + u2 + u3 + ..... + un + ...to = s.
(or)
n=1
un
=
s
(or)
simply un = s.
(b) Divergent series: If sn or - , the series said to be divergent.
(c) Oscillatory Series: If sn does not tend to a unique limit either finite or infinite it
is said to be an Oscillatory Series.
Note: Divergent or Oscillatory series are sometimes called non convergent series.
1.2.2 Geometric Series
The series, 1 + x + x2 + .....xn-1 + ... is (i) Convergent when x < 1, and its sum is 1 1- x (ii) Divergent when x 1 .
(iii) Oscillates finitely when x = -1 and oscillates infinitely when x < -1.
Proof: The given series is a geometric series with common ratio `x'
sn
= 1- xn 1- x
when x 1 [By actual division ? verify]
................
................
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