Informative line

Definitions

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

Formula for General Term of Sequence

  • 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 

                a1,a2,a3,_ _ _ _ an,_ _ _ _

                a1 is called the first term of the sequence 

                a2 is called the second term of the sequence....

  • We will use infinite sequences so each term an will always have a successor an+1.
  • Sequence can be defined as a function whose domain is the set of positive integer and an is like \(f(x)\), in sequences we use the symbol an for the nth 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.    \(\dfrac{3}{5}, \dfrac{-4}{25}, \dfrac{5}{125} , \dfrac{-6}{625},\dfrac{7}{3125}....(-1)^{n-1}\dfrac{n+2}{5^n}...\)    

Illustration Questions

Write down the first five terms of the sequence \(a_n = \dfrac{2^n}{1+3^{n+1}}\) .  

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

×

If \(a_n\) = some expression in n, just put the values of n in the expression to find the terms. 

In this case \(a_n = \dfrac{2^{n}}{1+3^{n+1}}\)

\(\therefore\;\;a_1 = \dfrac{2^1}{1+3^2} = \dfrac{2}{10} = \dfrac{1}{5}\)

\(a_2 = \dfrac{2^2}{1+3^3} = \dfrac{4}{28} = \dfrac{1}{7}\)

\(a_3 = \dfrac{2^3}{1+3^4} = \dfrac{8}{82} = \dfrac{4}{41}\)

\(a_4 = \dfrac{2^4}{1+3^ 5} = \dfrac{16}{244} = \dfrac{4}{61}\)

\(a_5 = \dfrac{2^5}{1+3^6} = \dfrac{32}{1+729} = \dfrac{32}{730} = \dfrac{16}{365}\)

Write down the first five terms of the sequence \(a_n = \dfrac{2^n}{1+3^{n+1}}\) .  

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

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

Option C is Correct

Illustration Questions

Find a formula for the general term an of the sequence.  \(\dfrac {1}{2}, \dfrac{-4}{3}, \dfrac{9}{4}, \dfrac{-16}{5},\dfrac{25}{6}, .....\) Assuming that pattern of first few terms continues.   

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

×

Observe the pattern of numerator and denominator of the term and try to relate them to the term number.

In this case 

\(a_1 = \dfrac{1}{2}, a_2=\dfrac{-4}{3} , a_3= \dfrac{9}{4}, a_4= \dfrac{-16}{5}, a_5= \dfrac{25}{6}\)

Observe that the denominator starts with 2 in \(a_1\), and increases by 1, whereas numerator is a perfect square of consecutive numbers with alternate positive and negative signs.  

 

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

Find a formula for the general term an of the sequence.  \(\dfrac {1}{2}, \dfrac{-4}{3}, \dfrac{9}{4}, \dfrac{-16}{5},\dfrac{25}{6}, .....\) Assuming that pattern of first few terms continues.   

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

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

Option A is Correct

Finding Limit for Convergent and Divergent Sequence 

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. 

  • If \(\lim\limits _{n\to \infty }\) an 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 \(\lim\limits _{n \to \infty } \; a_n = L\).

Illustration Questions

Find the limit of the following convergent sequence.  \(a_n\; = \sqrt{\dfrac{5n+3}{2n+5}}\)

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

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

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

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

×

For a convergent sequence 

\(\lim\limits _{n\to \infty} \; a_n\; = L \)    where   L is a finite number, called the limit of the sequence .  

In this case 

\(L = \lim\limits _{n \to \infty } \sqrt{\dfrac{5n + 3}{2n + 5}}\)

\(= \lim\limits _{n\to \infty} \sqrt {\dfrac{5 + \dfrac{3}{n}}{2 + \dfrac{5}{n}}}\)  ( Divide Numerator & Denominator by n in the square root )

\(= \sqrt {\dfrac{5+0}{2+0}} \;\;\;=\;\;\;\sqrt{\dfrac{5}{2}}\)

\(\therefore \;\;L = \sqrt {\dfrac{5}{2}}\)

Find the limit of the following convergent sequence.  \(a_n\; = \sqrt{\dfrac{5n+3}{2n+5}}\)

A

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

.

B

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

C

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

D

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

Option A is Correct

Illustration Questions

Find the limit of the following convergent sequence.  \(a_n = ln(3n^2 + 1)\;- ln \; (n^2 + 4)\)  

A L = ln 2

B L = ln 3

C L = ln 5

D L = ln 6

×

For a convergent sequence 

\(\lim\limits _{n \to \infty} \; a_n\; = L \)    where   L is a finite number, called the limit of the sequence.   

In this case 

\(a_n = ln (3n^2 +1) - ln (n^2 +4)\)

\( = ln \left(\dfrac{3n^2 + 1}{n^2 +4}\right)\)        ( use ln a – ln b = \(ln\dfrac{a}{b}\) )

\(\therefore\) If \(a_n = f (n)\) then \(f(n) =ln \left(\dfrac{3n^2 + 1}{n^2 + 4}\right)\)

\(\therefore \; L = \lim\limits _{n\to \infty }\; ln \left(\dfrac{3n^2 + 1}{n^2 + 4}\right)\)

\(= \lim\limits _{n\to\infty } \; ln \left(\dfrac{3+\dfrac{1}{n^2}}{1+\dfrac{4}{n^2}}\right)\; \)

\(= ln \;\dfrac{3+0}{1+0}\\ = ln\; 3\)

Find the limit of the following convergent sequence.  \(a_n = ln(3n^2 + 1)\;- ln \; (n^2 + 4)\)  

A

L = ln 2

.

B

L = ln 3

C

L = ln 5

D

L = ln 6

Option B is Correct

Increasing and Decreasing Sequences 

  • 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  

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

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

Illustration Questions

Which of the following sequence is increasing ? 

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 is said to be increasing if \(a_n < a_{n+1}\) for all \(n \geq 1 \; (n \in N)\).

Check the options 

(a)     \(a_n = \dfrac{1}{5n+4}\)

        \(a_n < a_{n+1}\)

\(\Rightarrow \; \dfrac{1}{5n+4}\; < \dfrac{1}{5(n+1) +4}\)

or \(\dfrac{1}{5n+4}<\dfrac{1}{5n+9} \) or \(5n+ 4 > 5n +9 \)

or \(4>9\) which is not true.

\(\therefore \; a_n\) is a decreasing sequence .

(b)   \(a_n = (-3)^{n+1}\)

         \(a_n < a_{n+1}\)

\(\Rightarrow \;(-3) ^{n+1} < (-3) ^{n+2}\)

\(\Rightarrow \; (-1)^{n+1} × 3^{n+1}\;<\; (-1)^{n+2} × 3^{n+2}\)

or  \((-1)^{n+1} < (-1) ^{n+2} × 3\)

which may or may not be true according to n.

\(\therefore\; a_n \) is non monotonic .  

(c)  \(a_n = n + \dfrac{1}{n}\)

       \(a_n <a_{n+1}\)

\(\Rightarrow \; n+ \dfrac{1}{n } < n+1 + \;\dfrac{1}{n+1}\)

or    \(\dfrac{1}{n} < \;1 + \dfrac{1}{n+1} \; or \dfrac{n+1+1}{n+1} - \dfrac{1}{n} > 0\)

\(or \;\; \dfrac{n^2 + 2n -n -1}{n (n+1)} > 0 \;\; or \;\; \dfrac{n^2 + n -1}{n (n+1)} > 0\)

which is true.

\(\therefore\;\; a_n \) is an increasing sequence.

(d)    \(a_n = \dfrac{1}{2n+5}\)

         \(a_n<a_{n+1}\)

\(\Rightarrow \; \dfrac{1}{2n+5}<\dfrac{1}{2 (n+1) +5} \; or \; \dfrac{1}{2n+5} < \dfrac{1}{2n+7}\)

\(or\;\; 2n+5 > 2n+7\;\; or\;\; 5>7 \)  which is not true. 

\(\therefore \; \; a_n \) is a decreasing sequence.  

Which of the following sequence is increasing ? 

A

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

.

B

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

C

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

D

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

Option C is Correct

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