mathematics-class-11-unit-10-chapter-04-sequence-and-series-l-4-10-agdsxnxffr8

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Recall series </primary>
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- Given a sequence $\\{a _n\\}$, the expression $a_1+a_2+a_3+\ldots \ldots$ is called a Series

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

- Sequence ordered list of numbers 
- Series - sum

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<footer style = "font-size: 18px"> $\rightarrow$ Sequence & series lecture-4 $\rightarrow$ <span style = "color:blue"> Recall series </span> $\rightarrow$ Finite and infinite series $\rightarrow$ Notation </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Finite and infinite series </primary>
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- If $\\{a_n\\}$ is finite the corresponding Series is finite. If $\\{a_n\\}_{n=1}^{\infty}$ is infinite then corresponding series is infinite.  
  
* $a_1+a_2+\cdots+a_n \ \ \rightarrow$ finite series.
* $a_1+a_2+ \cdots \ \ \rightarrow $ infinite series

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<footer style = "font-size: 18px">Sequence & series lecture-4 $\rightarrow$ Recall series $\rightarrow$ <span style = "color:blue"> Finite and infinite series </span> $\rightarrow$ Notation $\rightarrow$ Sequence of partial sums </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Sequence of partial sums </primary>
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- Consider the Sequence $\\{a_ n\\}_{n=1}^{\infty}.$

- We construct a new sequence $\\{S_ n\\}_{n=1}^{\infty}$ as follows,

- $ S_1=a_1 $

- $ S_2=a_1+a_2 $

- $ S_3=a_1+a_2+a_3$

- $S_ n=a_ 1+a_ 2+\cdots+a_ n$ $ \quad \quad   =\sum_ {i=1}^n a_i$

$ \vdots$  
- The sequence ${S_n}$ is called sequence of Partial sums of the series $\sum_{n=1}^{\infty} a_n.$  
- $S_n \rightarrow n^{\text {th }}$ partial sum

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<footer style = "font-size: 18px">Finite and infinite series $\rightarrow$ Notation $\rightarrow$ <span style = "color:blue"> Sequence of partial sums </span> $\rightarrow$ Notes $\rightarrow$ Convergent series </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Convergent series </primary>
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- For a given sequence $\\{a_n\\}$ and corresponding series $\sum_{n=1}^{\infty} a_n$, construct sequence $\left(S_n\right)_{n=1}^{\infty}$, the sequence of partial sum.

- If $(S_n)^{\infty}$ is convergent, then we say that $\sum_{n=1}^{\infty} a_n$ is convergent.

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<footer style = "font-size: 18px">Sequence of partial sums $\rightarrow$ Notes $\rightarrow$ <span style = "color:blue"> Convergent series </span> $\rightarrow$ Divergent series $\rightarrow$ Convergence </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Convergence </primary>
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- If $\underset{n \rightarrow \infty}{\operatorname{Lt}} S_n=L$

- then $\sum_{n=1}^{\infty} a_n=L$  
  
- Sequence - what happens as you progress towards end.

- Series "Summable"

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

- $\sum a_n$  arises from the sequence $\left(\frac{1}{2^n}\right)_{n=0}^{\infty}$

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<footer style = "font-size: 18px">Convergent series $\rightarrow$ Divergent series $\rightarrow$ <span style = "color:blue"> Convergence </span> $\rightarrow$ Sequence of partial sum $\rightarrow$ Sequence of partial sum </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Sequence of partial sum </primary>
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- $\begin{aligned} S_3 & =1+\frac{1}{2}+\frac{1}{4} \\\  & =2-\frac{1}{4}\end{aligned}$

- $S_4=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=2-\frac{1}{8}$  
- $\vdots$

- $S_n=\frac{2^n-1}{2^{n-1}}$

- $=2-\frac{1}{2^{n-1}} \ \ n= 1,2,3,\cdot\cdot\cdot$

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<footer style = "font-size: 18px">Convergence $\rightarrow$ Sequence of partial sum $\rightarrow$ <span style = "color:blue"> Sequence of partial sum </span> $\rightarrow$ Sequence of partial sum $\rightarrow$ Sequence vs series </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Sequence of partial sum </primary>
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- $
 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+ \cdot\cdot\cdot
 $

- The sequence of partial sum $\left(S_n\right)_{n=1}^{\infty}$
- $
 S_n:=2-\frac{1}{2^{n-1}}
 $
- $\forall n \in \mathbb{N}$

- $\operatorname{Lt}_{n \rightarrow \infty} \quad S_n=2$  
- $\therefore 1+\frac{1}{2}+\frac{1}{4}+\cdots \ \ \text { is convergent }$  
  
- $\begin{array}{r}1+\frac{1}{2}+\frac{1}{4}+\cdots =2\end{array}$

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<footer style = "font-size: 18px">Sequence of partial sum $\rightarrow$ Sequence of partial sum $\rightarrow$ <span style = "color:blue"> Sequence of partial sum </span> $\rightarrow$ Sequence vs series $\rightarrow$ Arithmetic progression </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Arithmetic progression </primary>
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- A sequence in which difference between two consecutive terms remains the same called an Arithmetic Sequence or (AP)

- A sequence $ \\{a_ n \\}_{n=1}^{\infty}$ is called arithmetic- sequence or AP if,

- $a_{n+1}=a_n+d \quad \forall n \geqslant 1$
- where "$d$" is a real number.  
- Here " $d$ " is called common difference.

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<footer style = "font-size: 18px">Sequence vs series $\rightarrow$ Arithmetic progression $\rightarrow$ <span style = "color:blue"> Arithmetic progression </span> $\rightarrow$ Notes $\rightarrow$ Facts </footer>

### <Success style="font-size:25px"> Sequence And Series L-4 </Success>
### <primary style="font-size:24px"> Facts </primary>
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- If we add a constant to each term in an AP, the resulting Sequence is again an AP

- $ \left(a_n\right)_{n=1}^{\infty}$

- $a_1, a_2, a_3, \ldots$

- $\left(b_n\right)_{n=1}^{\infty} \quad b_n=a_n+d^{\prime}$

- $a_1+d^{\prime}, a_2+d^{\prime}, a_3+d^{\prime} \cdots $

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<footer style = "font-size: 18px">Arithmetic progression $\rightarrow$ Notes $\rightarrow$ <span style = "color:blue"> Facts </span> $\rightarrow$ Thank you $\rightarrow$  </footer>