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 ${S_{n}}$ .

Example: $S_{n} = {\frac{n}{n + 1} / n \in Z^{+}}$ is a sequence.

Boundedness of a Sequence

Bounded above and Bounded below Sequences

(i) Bounded above Sequence: A sequence ${S_{n}}$ is said to be bounded above if $\exists$ a real number $k$ such that

S_{n} \leq k \forall n \in Z^{+}

where $k$ is called the upper bound of ${S_{n}}$ .

Least Upper Bound or Supremum of the Sequence: It is the smallest number which is greater than all the terms of the sequence.

Example: ${S_{n}} = {\frac{1}{n} / n \in Z^{+}}$ is bounded above with 1 as supremum.

(ii) Bounded below Sequence: A sequence ${S_{n}}$ is said to be bounded below if $\exists$ a real number $k$ such that

S_{n} \geq k \forall n \in Z^{+}

where $k$ is called the lower bound of ${S_{n}}$ .

Greatest lower bound or Infimum of the Sequence: It is the largest number which is less than all the terms of the sequence.

Example: ${S_{n}} = {\frac{1}{n} / n \in Z^{+}}$ 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: ${S_{n}} = {(- 1)^{n} / n \in Z^{+}}$ is bounded above (1 as supremum) and below (-1 as infimum).

Example: ${\frac{1}{n} / n \in Z^{+}}$ 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: ${- n / n \in Z^{+}}$ is bounded above by 0 but not bounded below.

Example 2: ${n / n \in Z^{+}}$ is bounded below by 0 but not bounded above.

Example 3: ${S_{n}} = {(- 1)^{n} n / n \in Z^{+}}$ is neither bounded below nor bounded above.

Limit Of A Sequence

The limit of a sequence is the value that the $n^{t h}$ 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 $lim_{n \to \infty} a_{n} \leq lim_{n \to \infty} b_{n}$

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 $lim_{n \to \infty} a_{n} = lim_{n \to \infty} b_{n} = L$ , then $lim_{n \to \infty} c_{n} = L$

The following results hold true for limits of sequences:

1) $lim_{n \to \infty} (a_{n} \pm b_{n}) = lim_{n \to \infty} a_{n} \pm lim_{n \to \infty} b_{n}$

2) $lim_{n \to \infty} c a_{n} = c \cdot lim_{n \to \infty} a_{n}$

3) $lim_{n \to \infty} (a_{n} \cdot b_{n}) = (lim_{n \to \infty} a_{n}) \cdot (lim_{n \to \infty} b_{n})$

4) $lim_{n \to \infty} (\frac{a_{n}}{b_{n}}) = \frac{lim_{n \to \infty} a_{n}}{lim_{n \to \infty} b_{n}} provided lim_{n \to \infty} b_{n} \neq 0$

5) $lim_{n \to \infty} a_{n}^{p} = {[lim_{n \to \infty} a_{n}]}^{p}$

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 $x_{n + 1} = \sqrt{x_{n} + 20}, n = 1, 2, 3, \dots$ , show that the sequence $x_{1}, x_{2}, x_{3} . .$ is convergent.

[2017, 10M]

2) Two sequences ${x_{n}}$ and ${y_{n}}$ are defined inductively by the following: $x_{1} = \frac{1}{2}$ , $y_{1} = 1$ , $x_{n} = \sqrt{x_{n - 1} y_{n - 1}}$ , $n = 2, 3, 4, \dots \frac{1}{y_{n}} = \frac{1}{2} (\frac{1}{x_{n}} + \frac{1}{y_{n - 1}})$ , $n = 2, 3, 4, \dots$ . Prove that $x_{n - 1} < x_{n} < y_{n} < y_{n - 1}$ , $n = 2, 3, 4, \dots$ and deduce that both the sequences converge to the same limit where $\frac{1}{2} < l < 1$ .

[2016, 10M]

3) Define ${x_{n}}$ by $x_{1} = 5$ and $x_{n + 1} = \sqrt{4 + x_{n}}$ for $n > 1$ . Show that the sequence converges to $(\frac{1 + \sqrt{17}}{2})$ .

[2010, 12M]

4) Discuss the convergence of the sequence ${x_{n}}$ where $x_{n} = \frac{\sin (\frac{n π}{2})}{8}$ .

[2010, 12M]

5) Let $a$ be a positive real number and ${x_{n}}$ a sequence of rational numbers such that $lim_{n \to \infty} x_{n} = 0$ . Show that $lim_{n \to \infty} a^{x_{n}} = 1$ .

[2003, 12M]

6) If $lim_{n \to \infty} a_{n} = l$ , then prove that $lim_{n \to \infty} \frac{a_{1} + a_{2} + \dots a_{n}}{n} = l$ .

[2001, 12M]

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