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 \(\left\{S_{n}\right\}\).
Example: \(S_{n}=\left\{\dfrac{n}{n+1} / n \in \mathbb{Z}^{+}\right\}\) is a sequence.
Boundedness of a Sequence
Bounded above and Bounded below Sequences
(i) Bounded above Sequence: A sequence \(\left\{S_{n}\right\}\) is said to be bounded above if \(\exists\) a real number \(k\) such that
\[S_{n} \leq k \text{ } \forall \text{ } n \in \mathbb{Z}^{+}\]where \(k\) is called the upper bound of \(\left\{S_{n}\right\}\).
Least Upper Bound or Supremum of the Sequence: It is the smallest number which is greater than all the terms of the sequence.
Example: \(\left\{S_{n}\right\}=\left\{\dfrac{1}{n} / n \in \mathbb{Z}^{+}\right\}\) is bounded above with 1 as supremum.
(ii) Bounded below Sequence: A sequence \(\left\{S_{n}\right\}\) is said to be bounded below if \(\exists\) a real number \(k\) such that
\[S_{n} \geq k \text{ } \forall \text{ } n \in \mathbb{Z}^{+}\]where \(k\) is called the lower bound of \(\left\{S_{n}\right\}\).
Greatest lower bound or Infimum of the Sequence: It is the largest number which is less than all the terms of the sequence.
Example: \(\left\{S_{n}\right\}=\left\{\dfrac{1}{n} / n \in \mathbb{Z}^{+}\right\}\) is bounded below with 0 as infimum.
Bounded Sequence
A sequence is said to be bounded if it is bounded from above and below.
Example: \(\left\{S_{n}\right\}=\left\{(-1)^{n} / n \in \mathbb{Z}^{+}\right\}\) is bounded above (1 as supremum) and below (-1 as infimum).
Example: \(\left\{\dfrac{1}{n} / n \in \mathbb{Z}^{+}\right\}\) is bounded above (1 as supremum) and below (0 as infimum).
Unbounded Sequence
A sequence which is not bounded is called an unbounded sequence.
Example 1: \(\left\{-n / n \in \mathbb{Z}^{+}\right\}\) is bounded above by 0 but not bounded below.
Example 2: \(\left\{n / n \in \mathbb{Z}^{+}\right\}\) is bounded below by 0 but not bounded above.
Example 3: \(\left\{S_{n}\right\}=\left\{(-1)^{n} n / n \in \mathbb{Z}^{+}\right\}\) is neither bounded below nor bounded above.
Limit Of A Sequence
The limit of a sequence is the value that the \(n^{th}\) 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 \(a_{n} \leq b_{n}\) for all \(n\) greater than some \(N\), then \(\lim_{n \rightarrow \infty} a_{n} \leq \lim_{n \rightarrow \infty} b_{n}\)
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 \(a_{n} \leq c_{n} \leq b_{n}\) for all \(n>N\) and \(\lim_{n \rightarrow \infty} a_{n}=\lim_{n \rightarrow \infty} b_{n}=L\), then \(\lim_{n \rightarrow \infty} c_{n}=L\)
The following results hold true for limits of sequences:
1) \(\lim_{n \rightarrow \infty}\left(a_{n} \pm b_{n}\right)=\lim_{n \rightarrow \infty} a_{n} \pm \lim_{n \rightarrow \infty} b_{n}\)
2) \(\lim_{n \rightarrow \infty} c a_{n}=c \cdot \lim_{n \rightarrow \infty} a_{n}\)
3) \(\lim_{n \rightarrow \infty}\left(a_{n} \cdot b_{n}\right)=\left(\lim_{n \rightarrow \infty} a_{n}\right) \cdot\left(\lim_{n \rightarrow \infty} b_{n}\right)\)
4) \(\lim_{n \rightarrow \infty}\left(\dfrac{a_{n}}{b_{n}}\right)=\dfrac{\lim_{n \rightarrow \infty} a_{n}}{\lim_{n \rightarrow \infty} b_{n}} \text { provided } \lim_{n \rightarrow \infty} b_{n} \neq 0\)
5) \(\lim_{n \rightarrow \infty} a_{n}^{p}=\left[\lim_{n \rightarrow \infty} a_{n}\right]^{p}\)
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 \(x_{1}=2\) and \(x_{n+1}=\sqrt{x_{n}+20}, n=1,2,3, \ldots\), show that the sequence \(x_{1}, x_{2}, x_{3} . .\) is convergent.
[2017, 10M]
2) Two sequences \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) are defined inductively by the following:\(x_{1}=\dfrac{1}{2}\), \(y_{1}=1\), \(x_{n}=\sqrt{x_{n-1} y_{n-1}}\), \(n=2,3,4, \ldots \dfrac{1}{y_{n}}=\dfrac{1}{2}\left(\dfrac{1}{x_{n}}+\dfrac{1}{y_{n-1}}\right)\), \(n=2,3,4, \ldots\). Prove that \(x_{n-1} < x_{n} < y_{n} < y_{n-1}\), \(n=2,3,4, \ldots\) and deduce that both the sequences converge to the same limit where \(\dfrac{1}{2} < l < 1\).
[2016, 10M]
3) Define \(\left\{x_{n}\right\}\) by \(x_{1}=5\) and \(x_{n+1}=\sqrt{4+x_{n}}\) for \(n>1\). Show that the sequence converges to \(\left(\dfrac{1+\sqrt{17}}{2}\right)\).
[2010, 12M]
4) Discuss the convergence of the sequence \(\left\{x_{n}\right\}\) where \(x_{n}=\dfrac{\sin \left(\dfrac{n \pi}{2}\right)}{8}\).
[2010, 12M]
5) Let \(a\) be a positive real number and \(\left\{x_{n}\right\}\) a sequence of rational numbers such that \(\lim _{n \rightarrow \infty} x_{n}=0\). Show that \(\lim _{n \rightarrow \infty} a^{x_n}=1\).
[2003, 12M]
6) If \(\lim _{n \rightarrow \infty} a_{n}=l\), then prove that \(\lim _{n \rightarrow \infty} \dfrac{a_{1}+a_{2}+\ldots a_{n}}{n}=l\).
[2001, 12M]