Introduction : A sequence is a function f whose domain is a set of natural numbers N and range is a set of real numbers R.
\[f:N\rightarrow R\]
terms of a sequence are denoted as \((x_n)_{n=1}^{\infty} =\{ x_1, x_2 , x_3 ,……,x_n, ……..\}\)
Example 1
\((n)_{n=1}^{\infty}=\{1,2,3,4,………,n,….\}\)\((\frac{1}{n})_{n=1}^{\infty}=\{1,\frac{1}{2},\frac{1}{3},……,\frac{1}{n},…..\}\)\(((-1)^{n+1})_{n=1}^{\infty}=\{1,-1,1,-1,………,(-1)^{n+1},…..\}\)\((\frac{1}{2^n})_{n=1}^{\infty}=\{\frac{1}{2},\frac{1}{2^2},\frac{1}{2^3},……,\frac{1}{2^n},…..\}\)
Example 2 :
Fibonacci Sequence :
Fibonacci Sequence is a sequence of numbers where general term is the sum of the two preceding numbers.
To form Fibonacci sequence, Lets take first two numbers as
first term \[a_0 =1\]
second term \[a_1 =1\]
now third term a2 is obtained as
\[a_2 =a_0+a_1\]
=1+1
\[a_2=2\]
fourth term \[a_3\]
\[a_3 =a_1+a_2\]
=1+2
\[a3=3\]
fifth term \[a_4\]
\[a_4=a_3+a_2\]
and so on
and hence the general term of the Fibonacci Sequence is
\[a_n=a_{n-2}+a_{n-1}\]
Thus we obtained Fibonacci Sequence as
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
Constant Sequence
If all the terms of sequence are equal then it is called constant.
For example \((\sin(n\pi))_{n=1}^{\infty}=\{0,0,0,0,………,0,…..\}\)
Oscillatory Sequence
If terms of a sequence are alternatively equal then it is oscillatory.
For example \((\cos(n\pi))_{n=1}^{\infty}=((-1)^{n})_{n=1}^{\infty}=\{-1,1,-1,1,………,(-1)^{n},…..\}\)
Finite Sequence
If the terms of a sequence ends after some point then it is finite. A finite sequence is also called a list.
For examples \(\{0,1\}\) and \(\{-10,-8,-6,-4,-2,0,2,4,6,8,10\}\) are finite sequences.
Infinite Sequence
If the terms of the sequence never ends that is goes on infinitely either towards left or right or both direction in the set of reals then it is an infinite sequence.
For examples set of negative even integers \( =\{….,-2n,…,-8,-6-4,2,0,2,4,6,….,2n,….\}\) and set of positive odd integers \(\{1,3,5,7,….2n-1,…..\}\)
Increasing Sequence
If the terms of a sequence are increasing
Decreasing Sequence
Bounded Sequence
Unbounded Sequence
class=”has-inline-color has-luminous-vivid-orange-color”>Question (24) : Let \(A(n) = \int\limits_n^{n + 1} {\frac{1}{{{x^3}}}} dx\) for \(n\ge 1\) for \(c\in R\) let \({\lim _{n \to \infty }}{n^c}A(n) = L\) then \[\]
(A) \(L =0\) if \( c> 3\) (B) \(L =1\) if \( c= 3\) (C) \(L =2\) if \( c= 3\) (D) \(L =\infty\) if \( 0<c< 3\)
class=”has-inline-color has-luminous-vivid-amber-color”>Solution:
\[A(n) = \int\limits_n^{n + 1} {\frac{1}{{{x^3}}}} dx\]
\[= \left[-\frac{1}{2x^2}\right]_{n}^{n + 1}\]
\[= -\frac{1}{2}\left[{\frac{1}{{{(n + 1)^2}}}}-{\frac{1}{{{n^2}}}}\right]\]
\[=-\frac{1}{2}\frac{n^2-(n+1)^2}{n^2(n+1)^2}\]
\[=-\frac{1}{2}\frac{n^2-n^2-1-2n}{n^2(n+1)^2}\]
\[=-\frac{1}{2}\frac{-1-2n}{n^2(n+1)^2}\]
\[=\frac{1}{2n^2(n+1)^2}+\frac{1}{n(n+1)^2}\]
\[\Rightarrow A(n) =\frac{1}{2n^4(1+\frac{1}{n})^2}+\frac{1}{n^3(1+\frac{1}{n})^2}\]
Now \(L ={\lim _{n \to \infty }}{n^c}A(n) ={\lim _{n \to \infty }}{n^c}\left[\frac{1}{2n^4(1+\frac{1}{n})^2}+\frac{1}{n^3(1+\frac{1}{n})^2}\right] \)
\[L ={\lim _{n \to \infty }}{n^{c-4}}\frac{1}{2(1+\frac{1}{n})^2}+{\lim _{n \to \infty }}\frac{n^{c-3}}{(1+\frac{1}{n})^2}\]
Clearly if \(c=3\) then \( L =0+1 = 1\)
Hence option \((B) L =1\) if \( c= 3\) is correct