2.10 Infinite Sequences

Definition 2.10.1.

An infinite sequence \(\{a_n\}^{\infty }_{n=1} = a_1, a_2,\cdots \cdots \cdots \hspace {0.3cm}\) is a function on \(n\) whose domain is the set of positive integers.

Definition 2.10.2.

A sequence is bounded if \(\exists \) numbers, \(p\) and \(q\) \(\ni p \leq a_n \leq q \hspace {0.2cm}\forall \) values of \(n\in \mathbb {Z}^+\).

Example 2.10.3.

The sequence \(\{a_n\}^{\infty }_{n=1} = \dfrac {3}{2}, \dfrac {5}{4},\dfrac {7}{6}, \cdots \cdots \cdots , \dfrac {2n+ 1}{2n}\), is bounded since, \(\hspace {0.2cm} \forall n \in \mathbb {Z}^+\hspace {0.4cm} 1 \leq a_n \leq 2\).

Definition 2.10.4.

A sequence \(\{a_n\}^{\infty }_{n=1}\) is said to be increasing if \(\hspace {0.4cm} a_1\leq a_2\leq a_3\cdots \cdots \cdots \leq a_n \leq \cdots \cdots \)

A sequence is decreasing if \(\hspace {0.4cm} a_1\geq a_2\geq a_3\cdots \cdots \cdots \geq a_n \geq \cdots \cdots \)

Definition 2.10.5.

A sequence \(\{a_n\}^{\infty }_{n=1}\) is said to converge to a finite number \(a\) if \(\hspace {0.4cm} \lim \limits _{n\rightarrow \infty } a_n = a\).

A sequence which converges is called a convergent sequence otherwise it is a divergent sequence.

Some Properties of Sequences

1.

Every bounded increasing sequence (or decreasing sequence) converges.

2.

Every unbounded sequence diverges.

3.

A convergent (divergent) sequence remains convergent (divergent) after any or all its terms altered.

4.

Let \(\lim \limits _{n \rightarrow \infty } a_n = a\) and \(\lim \limits _{n \rightarrow \infty } b_n = b\)

(a): \(\lim \limits _{n \rightarrow \infty } K a_n = Ka\)
(b): \(\lim \limits _{n \rightarrow \infty }( a_n\pm b_n) = \lim \limits _{n \rightarrow \infty } a_n \pm \lim \limits _{n \rightarrow \infty } b_n = a \pm b\)
(c): \(\lim \limits _{n \rightarrow \infty } (a_n\cdot b_n) = \lim \limits _{n \rightarrow \infty } a_n \cdot \lim \limits _{n \rightarrow \infty } b_n = a\cdot \)
(d): \(\lim \limits _{n \rightarrow \infty } \dfrac {a_n}{b_n} = \dfrac { \lim \limits _{n \rightarrow \infty } a_n}{\lim \limits _{n \rightarrow \infty } b_n} = \dfrac {a}{b}\) provided \(b \neq 0\).

5.

If \(\{a_n\}^{\infty }_{n=1}\) is a sequence of nonzero terms and if \(\lim \limits _{n \rightarrow \infty } a_n = \infty \) then \(\lim \limits _{n \rightarrow \infty }\dfrac {1}{a_n} = 0\)

6.

If \(r>1\), then \(\lim \limits _{n \rightarrow \infty } r^n = +\infty \)

7.

If \(|r|<1\), then \(\lim \limits _{n \rightarrow \infty } r^n = 0\)

Example 2.10.6.

Let a sequence be \(\Big \{\Big (\dfrac {1}{2}\Big )^n\Big \}\). Then \(\lim \limits _{n \rightarrow \infty } \Bigg (\dfrac {1}{2}\Bigg )^n = 0\).

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.