mathematics-class-11-unit-10-chapter-03-sequence-and-series-l-3

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Recall </primary>
<hr>
<main>

- **Sequence**

- $f : S \longrightarrow {R}$
   - $ S \subseteq \mathbb{N} \cup \\{0\\} $
   - $ \\{a_n\\}_{n=1}^{\infty} \quad a_n=f(n)$
- $(a_n)_{n=1}^{\infty} \text { or } $  
  - $ \\{a_1, a_2, a_3, \ldots a_n \cdot \ldots \\} $
- Closed form expression

- $\\{a_ n\\}_ {n=1}^{\infty} \quad a_n=\frac{1}{n^2} \forall n \geqslant N $

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Sequence & series lecture-3 $\rightarrow$ <span style = "color:blue"> Recall </span> $\rightarrow$ Recall $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- **1)** $(\frac{1}{n^2})_{n=1}^{\infty}$

- $ 1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16} \cdots \frac{1}{n^2}, \cdots $

- $ \underset{n\rightarrow\infty}{Lim} \frac{1}{n^2}=0$

- **2)** $\\{(-1)^n\\}_{n-2}^{\infty}$

- $1, -1, 1, -1, ........$

- **3)** $\\{n^2\\}_{n=1}^{\infty}$

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Recall $\rightarrow$ Recall $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Series $\rightarrow$ Remark </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Question </primary>
<hr>
<main>

- Finding Sum of infintely many real numbers.
  
    
      
- **Decimal expansion**

- $\frac{10}{3} = 3.333...........$

- $3 + \frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \cdots = \frac{10}{3}$

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Series $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Question </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- **Observation**

- $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+ \cdots = 1 $

- $1+2+3+4+ \cdots$ cannot be finite.

- $1 + (\frac{-1}{2}) + ( \frac{1}{3}) + (-\frac{1}{4}) + \cdots $

- $[1+(\frac{-1}{2})] + [\frac{1}{3} + (-\frac{1}{4})] + \cdots$

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Question $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Notation </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- **Suppose we rearrange the terms of given series as follows:**

- $(-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}-\frac{1}{8}+1)$

- $(-\frac{1}{10}-\frac{1}{12}-\frac{1}{14}-\frac{1}{16}-\frac{1}{18} + \frac{1}{3})$

- $1+(-\frac{1}{2})+(\frac{1}{3})+(-\frac{1}{4}) +\cdots $

- $[1+(-\frac{1}{2})]+[\frac{1}{3}+(-\frac{1}{4})] + \cdots $

</div>
<div style='float: right; width:0%;'>

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

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Notation </primary>
<hr>
<main>

- $a_1, a_2 \cdots a_n$

- $a_1+a_2+a_3 \cdots + a_n$

- Sigma notation, $\sum$
- $a_1+a_2+ \cdots +a_n = \underset {i=1} {\overset{n} \sum} a_i$  
    - $ i-\text { index of summation } $
    - $ a_1+a_2+ \cdots + a_n $

- $ = \underset{i=1}{\overset{n}\sum} a_i = \underset{j=1}{\overset{n}\sum} a_j = \underset{r=1}{\overset{n}\sum} a_r$

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Notation </span> $\rightarrow$ Notation $\rightarrow$ Question </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Question </primary>
<hr>
<main>

- Find $\underset{r=1}{\overset{8} \sum} (-1)^r$  
  
- **Solution:**

- $ \underset{r=1}{\overset{8} \sum} (-1)^r = (-1+1)+(-1+1)+(-1+1)+(-1+1)$

- $\quad \quad  = 0$

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Notation $\rightarrow$ Question $\rightarrow$ <span style = "color:blue"> Question </span> $\rightarrow$ Series $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Sequence And Series L-3 </Success>
### <primary style="font-size:24px"> Series </primary>
<hr>
<main>

- Given a sequence $\\{a_n\\}$, the expression,  $a_1+a_2+a_3+ \cdots $

- is called a series.

- associated with the sequence $\\{a_n\\}.$

- Sequence -ordered list of numbers 
- Series - sum

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Question $\rightarrow$ Question $\rightarrow$ <span style = "color:blue"> Series </span> $\rightarrow$ Thank you $\rightarrow$  </footer>