Learn definition of convergence and divergence, monotonic & limit of a sequence. Practice sequences definition and series calculus.

- Sequence is a list of number written in a definite order or having a particular pattern.
- e.g 2,4,6,8,_ _ _ _

It is usually written as

a_{1},a_{2},a_{3},_ _ _ _ a_{n,}_ _ _ _

a_{1} is called the first term of the sequence

a_{2} is called the second term of the sequence....

- We will use infinite sequences so each term a
_{n}will always have a successor a_{n+1}. - Sequence can be defined as a function whose domain is the set of positive integer and a
_{n}is like \(f(x)\), in sequences we use the symbol a_{n}for the n^{th }term, instead of \(f(x)\). - There are three different ways of representing a sequence.

(1) By using the notation \(\{a_n\}^{\infty}_{n = 1}\)

For e.g. the sequence \(\left\{\dfrac{2n}{5n+1}\right\}^{\infty}_{n=1}\) will mean the sequence \(\left\{\dfrac{2}{6} , \dfrac{4}{11} \dfrac{6}{16},....\dfrac{2n}{5n + 1}...\right\}\).

(2) By using the defining formula

e.g. \(a_n= \dfrac{n}{n+3}\) will mean the sequence \(\dfrac{1}{4},\dfrac{2}{5}, \dfrac{n}{n+3}...\)

(3) By writing the terms of the sequence

e.g. \(\dfrac{3}{5}, \dfrac{-4}{25}, \dfrac{5}{125} , \dfrac{-6}{625},\dfrac{7}{3125}....(-1)^{n-1}\dfrac{n+2}{5^n}...\)

A \(\dfrac{1}{5}, \dfrac{4}{9}, \dfrac{12}{5}, \dfrac{11}{27},\dfrac{19}{352}\)

B \(\dfrac{1}{2}, \dfrac{4}{7}, \dfrac{15}{11}, \dfrac{19}{23}, \dfrac{521}{32}\)

C \(\dfrac{1}{5}, \dfrac{1}{7}, \dfrac{4}{41}, \dfrac{4}{61}, \dfrac{16}{365}\)

D \(\dfrac{1}{4}, \dfrac{5}{7}, \dfrac{9}{13}, \dfrac{212}{51}, \dfrac{567}{93}\)

A \(a_n=\dfrac{(-1)^{n+1}\;n^2}{n+1}\)

B \(a_n=\dfrac{(-1)^{n}\;n^2}{n+1}\)

C \(a_n=\dfrac{n+2}{n^2}\)

D \(\dfrac{n^2}{n+2}\)

A sequence \(\{a_n\}\) has the limit L and we write \(\lim\limits_{n\to \infty \; }a_n = L \;\;\text{or}\;\;a_n \to L\) as \(n \to \infty \) if we

can make the term a_{n} as close to L as we like by taking n sufficiently large.

- If \(\lim\limits _{n\to \infty }\) a
_{n}exists or it is a finite number we say that sequence converges ( or is convergent ), otherwise we say the sequence diverges ( or is divergent ).

Pattern of a converging sequence when \(\lim\limits_{n\to\infty}\;\;a_n = L\)

Pattern of a divergent sequence when \(\lim\limits_{n \to \infty }\; a_n \) does not exist.

- If \(\lim \limits _{n\to \infty }\; f (n) = L \) and \(f (n) = a_n \) when n is an integer, then \(\lim\limits _{n \to \infty } \; a_n = L\).

A \(L = \sqrt {\dfrac{5}{2}}\)

B \(L = \dfrac{5}{2}\)

C \(L= \dfrac{1}{2}\)

D \(L = \sqrt {\dfrac{5}{3}}\)

A L = ln 2

B L = ln 3

C L = ln 5

D L = ln 6

- A sequence \(\{a_n\}\) is called increasing if \(a_n<a_{ n+1}\) ; for all \(n\geq1\), that is \(a_1< a_ 2<a_3 .......a_n,\) where as it is called decreasing if \(a_n > a_{n+1}\) for all \(n\geq 1\).
- A sequence is said to be monotonic if it is either increasing or decreasing .
- To show that a sequence \(\{a_n\}\)is increasing point \(a_n<a_{n+1} \;\forall \;n\)
- And to show that it is decreasing point

- e.g \(a_n = \dfrac{5}{n+4}\) is a decreasing sequence as

\(\dfrac{5}{n+4} > \dfrac{5}{(n+1)+4}\; or \; a_n>a_{n +1}\)

\(\dfrac{5}{n+4} > \dfrac{5}{n+5}\)

A \(a_n = \dfrac{1}{5n+4}\)

B \(a_n = (-3)^{n+1}\)

C \(a_n = n +\dfrac{1}{n}\)

D \(a_n= \dfrac{1}{2n+5}\)