Learn definition of convergence and divergence, monotonic & limit of a sequence. Practice sequences definition and series calculus.
It is usually written as
a1,a2,a3,_ _ _ _ an,_ _ _ _
a1 is called the first term of the sequence
a2 is called the second term of the 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 \(\dfrac{n^2}{n+2}\)
\(\dfrac{5}{n+4} > \dfrac{5}{(n+1)+4}\; or \; a_n>a_{n +1}\)
\(\dfrac{5}{n+4} > \dfrac{5}{n+5}\)
A \(a_n = \dfrac{1}{5n+4}\)
B \(a_n = (-3)^{n+1}\)
C \(a_n = n +\dfrac{1}{n}\)
D \(a_n= \dfrac{1}{2n+5}\)
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 an as close to L as we like by taking n sufficiently large.
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.
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