Sequence
We will cover following topics
Definition
Sequence is an enumerated collection of objects. Repetitions are allowed and order matters in a sequence.
A sequence is denoted by {Sn}.
Example: Sn={nn+1/n∈Z+} is a sequence.
Boundedness of a Sequence
Bounded above and Bounded below Sequences
(i) Bounded above Sequence: A sequence {Sn} is said to be bounded above if ∃ a real number k such that
Sn≤k ∀ n∈Z+where k is called the upper bound of {Sn}.
Least Upper Bound or Supremum of the Sequence: It is the smallest number which is greater than all the terms of the sequence.
Example: {Sn}={1n/n∈Z+} is bounded above with 1 as supremum.
(ii) Bounded below Sequence: A sequence {Sn} is said to be bounded below if ∃ a real number k such that
Sn≥k ∀ n∈Z+where k is called the lower bound of {Sn}.
Greatest lower bound or Infimum of the Sequence: It is the largest number which is less than all the terms of the sequence.
Example: {Sn}={1n/n∈Z+} is bounded below with 0 as infimum.
Limit Of A Sequence
The limit of a sequence is the value that the nth term of the sequence tends to as n approaches infinity.
If such a limit exists, the sequence is said to be convergent, otherwise it called divergent.
Some properties of the limits of sequences are given below:
1) The limit of a sequence is unique.
2) If an≤bn for all n greater than some N, then limn→∞an≤limn→∞bn
3) If a sequence is bounded and monotonic, then it is convergent.
4) A sequence is convergent if and only if its every subsequence is convergent.
3) Squeeze Theorem: If an≤cn≤bn for all n>N and limn→∞an=limn→∞bn=L, then limn→∞cn=L
The following results hold true for limits of sequences:
1) limn→∞(an±bn)=limn→∞an±limn→∞bn
2) limn→∞can=c⋅limn→∞an
3) limn→∞(an⋅bn)=(limn→∞an)⋅(limn→∞bn)
4) limn→∞(anbn)=limn→∞anlimn→∞bn provided limn→∞bn≠0
5) limn→∞apn=[limn→∞an]p
Cauchy Sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses.
PYQs
Limit of a Sequence
1) Let x1=2 and xn+1=√xn+20,n=1,2,3,…, show that the sequence x1,x2,x3.. is convergent.
[2017, 10M]
2) Two sequences {xn} and {yn} are defined inductively by the following:x1=12, y1=1, xn=√xn−1yn−1, n=2,3,4,…1yn=12(1xn+1yn−1), n=2,3,4,…. Prove that xn−1<xn<yn<yn−1, n=2,3,4,… and deduce that both the sequences converge to the same limit where 12<l<1.
[2016, 10M]
3) Define {xn} by x1=5 and xn+1=√4+xn for n>1. Show that the sequence converges to (1+√172).
[2010, 12M]
4) Discuss the convergence of the sequence {xn} where xn=sin(nπ2)8.
[2010, 12M]
5) Let a be a positive real number and {xn} a sequence of rational numbers such that limn→∞xn=0. Show that limn→∞axn=1.
[2003, 12M]
6) If limn→∞an=l, then prove that limn→∞a1+a2+…ann=l.
[2001, 12M]