mathematics-class-11-unit-10-chapter-02-sequence-and-series-l-2-10-1mmhw44ykxo

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Graphs for sequences </primary>
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- consider the sequence

- $
\\{a_ n\\}_{n=1}^{\infty}
 $

- Consider $\\{a_ n\\}_ {n=1}^{\infty}; \\ a_n=\sqrt{n}$

- Consider $\\{b_ n\\}_ {n=1}^{\infty} \\ b_n=\frac{1}{n}$

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<footer style = "font-size: 18px">Sequence & series lecture - 2 $\rightarrow$ Recall $\rightarrow$ <span style = "color:blue"> Graphs for sequences </span> $\rightarrow$ Graphs for sequences $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Write first five terms of the sequence given by the formula,
- $a_n=n(n+2)$

- **Solution:**    
- $\begin{aligned} & a_1=1(1+2)=3 \\\ & a_2=2(2+2)=2 \cdot 4=8 \\\ & a_3=3(3+2)=3 \cdot 5=15 \\\ & a_4=4(4+2)=4 \cdot 6=24 \\\ & a_5=5(5+2)=5 \cdot 7=35\end{aligned}$

- $\\{3,8,15,24,35, \cdots , \cdots, n(n+2), \cdots \\}$

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<footer style = "font-size: 18px">Graphs for sequences $\rightarrow$ Graphs for sequences $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Find first 4 terms of the sequence given by $a_n=n \cdot \frac{n^2+5}{4}$  
  
- **Solution:**

- $\begin{aligned} & a_1=1 \cdot \frac{1^2+5}{4}=1 \cdot \frac{6}{4} =\frac{3}{2}  \\\ & \begin{aligned} a_2 & =2 \cdot \frac{2^2+5}{4}=2 \cdot \frac{4+5}{4} \\\ & =2 \cdot \frac{9}{4}=\frac{9}{2}\end{aligned} \\\ & a_3=3 \cdot \frac{3^2+5}{4}\end{aligned}$

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<footer style = "font-size: 18px">Graphs for sequences $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Solution $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $a_3=3 \cdot \frac{9+5}{4}$  
- $\ =3 \cdot \frac{14}{4}$  
- $\ =3 \cdot \frac{7}{2} =\frac{21}{2}$

- $a_4  =4 \cdot \frac{4^2+5}{4}$  
- $\ =4 \cdot \frac{16+5}{4}$

- $\ =4 \cdot \frac{21}{4}$  
- $ \=21$

- $\frac{3}{2}, \frac{9}{2}, \frac{21}{2}, 21 $

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Find $9^{\text {th }}$ term of the sequence $\\{a_n\\} _{n=1}^{\infty}$

- $
 a_n=(-1)^{n-1} \cdot n^3
 $  
  
- **Solution:**

- $a_9=(-1)^{9-1} \cdot 9^3$
- $\begin{aligned} & =(-1)^8 \cdot 9^3 \\ & =81 \times 9 \\ & =729\end{aligned}$

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### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- First three terms of the sequence $\\{a_n\\} _{n=1}^{\infty}$

- $a_ n = a_{n-1} - 1, n > 2 $  
- $a_1=a_2=2$
  
    
- **Solution:**

- $\begin{aligned} & a_1=2, a_2=2 \text { given } \\\ & a_3=a_{3-1}-1=a_2-1 \\\ & =2-1 \\\ & =1 \\\ & \end{aligned}$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Find first 4 terms of the sequence $\\{a_n\\} _{n=1}^{\infty}$, where

- $\begin{aligned}& a_1=3 \\\ & a_n=3 a_{n-1} \quad n \geqslant 2 \end{aligned}$  
  
    
- **Solution:**

- $\begin{aligned} & a_1=3 \quad \text { given } \\\ & a_2=3 a_1=3 \cdot 3=3^2 \\\ & a_3=3 a_2=3 \cdot 3^2=3^3 \\\ & a_4=3 a_3=3 \cdot 3^3=3^4\end{aligned}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Solution $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Consider the sequence $ \\{a_n\\}_{n=i}^{\infty}$ where $\quad a_n=\frac{1}{n}$
- $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$

- {$0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \cdots \frac{n-1}{n}, \cdots$}
 
- {$0, \frac{1}{2}, \frac{2}{3}, \cdots 1-\frac{1}{n}, \cdots.$}

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<footer style = "font-size: 18px">Example $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $ \\{\sqrt{n}\\}_{n=1}^{\infty}$

- $\\{1, \sqrt{2}, \sqrt{3}, \cdots \sqrt{n}, \cdots \\} $

- $\begin{aligned} & a_{10001}=\sqrt{10001} \\\  &>100 . \\\  & a_m>k\end{aligned}$

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### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $ \\{1,-1,1,-1, \cdots(-1)^{n+1}, \cdots\\}$

- Consider the sequence $\\{a_n \\}_{n=i}^{\infty}$

- where  $ a_n = \frac{1}{n} $

- 1, $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$,

- $\{0, \frac{1}{2}, \frac{2}{3}, \cdots . .1- \frac{1}{n}\},.$.

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### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Convergent sequence </primary>
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- A sequence $\\{a_n \\} _{n=1}^{\infty}$ is said to be convergent if as " $n$ " increases $a_n$ 's become sofficiently close to a number $"L".$

- $L$ is called limit of the sequence.

- $ \operatorname{Lt}_ {n \rightarrow \infty} a_n=L$

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### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Consider the
- sequence $(a_n)_{n=1}^{\infty}$

- $a_n=\frac{5}{n^2}$

- $\\{a_ n\\}_{n=1}^{\infty}$

- $5, \frac{5}{2^2}, \frac{5}{3^2}, \frac{5}{4^2} \cdots$

- $\operatorname{Lt}_{n \rightarrow \infty} \frac{5}{n^2}=0$

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### <Success style="font-size:25px"> Sequence And Series L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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 $ \left\{a_n\right\}_{n=1}^{\infty}$

- $ a_n=\frac{4-7 n^6}{n^6+3}$

- $ a_n=\frac{n^6\left(\frac{4}{n^6}-7\right)}{n^6\left(1+\frac{3}{n^6}\right)}$

- $\begin{aligned} a_n & =\frac{\frac{4}{n^6}-7}{1+\frac{3}{n^6}} \\\\ & \operatorname{Lt}_{n \rightarrow \infty} a_n & =\frac{-7}{1}=-7\end{aligned}$

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