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- Sequence and Series

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Get Started Now- Intro Lesson: a5:11
- Intro Lesson: b6:15
- Lesson: 1a2:27
- Lesson: 1b2:03
- Lesson: 1c2:21
- Lesson: 2a2:53
- Lesson: 2b5:16
- Lesson: 2c5:09
- Lesson: 3a4:01
- Lesson: 3b2:10
- Lesson: 3c3:11

In this lesson, we will talk about what sequences are and how to formally write them. Then we will learn how to write the terms out of the sequences when given the general term. We will also learn how to write the general term when given a sequence. After learning the notations of sequences, we will take a look at the limits of sequences. Then we will take a look at some definitions and properties which will help us take the limits of complicating sequences. These theorems include the squeeze theorem, absolute value sequences, and geometric sequences.

Note:

1. If a sequence has the limit $L$, then we can say that:

$\lim$_{n →$\infty$} $a$_{$n$}$=L$

If the limit is finite, then it is convergent. Otherwise, it is divergent.

2. If the limit of the sequences {$a_n$} and {$b_n$} are finite and $c$ is constant, then we can say that

i) $\lim$_{n →$\infty$} $(a_n+b_n)=\lim$_{n →$\infty$} $a_n+$$\lim$_{n →$\infty$} $b_n$.

ii) $\lim$_{n →$\infty$} $(a_n-b_n)=\lim$_{n →$\infty$} $a_n-$$\lim$_{n →$\infty$} $b_n$.

iii) $\lim$_{n →$\infty$} $ca_n=c$ $\lim$_{n →$\infty$} $a_n$.

iv) $\lim$_{n →$\infty$}$(a_nb_n)=$ $\lim$_{n →$\infty$}$a_n*$ $\lim$_{n →$\infty$} $b_n$.

v) $\lim$_{n →$\infty$} $[a_n$$\div$$b_n]$
$=\lim$_{n →$\infty$}$a_n$$\div$
$\lim$_{n →$\infty$}$b_n$$,$$b_n\neq0$.

3. If $a_n\leq c_n\leq b_n$ and $\lim$_{n →$\infty$} $a_n=$ $\lim$_{n →$\infty$} $b_n=L$, then $\lim$_{n →$\infty$} $c_n=L$.

4.if $\lim$_{n →$\infty$} $|a_n|=0$, then $\lim$_{n →$\infty$} $a_n=0$ as well.

5. We say that:

Where the sequence {$x^n$} is convergent for -1< $x \leq$ 1, and divergent if $x$ > 1.

1. If a sequence has the limit $L$, then we can say that:

$\lim$

If the limit is finite, then it is convergent. Otherwise, it is divergent.

2. If the limit of the sequences {$a_n$} and {$b_n$} are finite and $c$ is constant, then we can say that

i) $\lim$

ii) $\lim$

iii) $\lim$

iv) $\lim$

v) $\lim$

3. If $a_n\leq c_n\leq b_n$ and $\lim$

4.if $\lim$

5. We say that:

Where the sequence {$x^n$} is convergent for -1< $x \leq$ 1, and divergent if $x$ > 1.

- IntroductionOverview:a)Notation of Sequencesb)Definitions and theorems of Sequences
- 1.
**Finding the terms of a sequence**

Find the first five terms of the following sequences.a)$a_n=3(-1)^n$b)$a_n$= $\frac{n+1}{\sqrt{n+1}}$c){$cos(\frac{n\pi}{2})$} - 2.
**Finding the formula for a sequence**

Find the formula for the general term $a_n$ for the following sequencesa){$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, ...$}b){$\frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, ...$}c){-1, 4, -9, 16, ... } - 3.
**Convergence and divergence of sequences**

Evaluate the limits and determine if the following limits are converging or diverging.a)$\lim$_{n →$\infty$}$\frac{(-1)^n}{n^2}$b)$\lim$_{n →$\infty$}$6(\frac{1}{2})^n$c)$\lim$_{n →$\infty$}$\frac{n^3+n+1}{n^2+1}$

8.

Sequence and Series

8.1

Introduction to sequences

8.2

Introduction to infinite series

8.3

Convergence and divergence of normal infinite series

8.4

Convergence and divergence of geometric series

8.5

Divergence of harmonic series

8.6

P Series

8.7

Alternating series test

8.8

Divergence test

8.9

Comparison and limit comparison test

8.10

Integral test

8.11

Ratio test

8.12

Absolute and conditional convergence

8.13

Radius and interval of convergence with power series

8.14

Functions expressed as power series

8.15

Taylor and maclaurin series

8.16

Approximating functions with Taylor polynomials and error bounds

We have over 320 practice questions in AP Calculus BC for you to master.

Get Started Now8.1

Introduction to sequences

8.3

Convergence and divergence of normal infinite series

8.4

Convergence and divergence of geometric series

8.5

Divergence of harmonic series

8.6

P Series

8.7

Alternating series test

8.8

Divergence test

8.9

Comparison and limit comparison test

8.10

Integral test

8.11

Ratio test

8.12

Absolute and conditional convergence

8.13

Radius and interval of convergence with power series

8.14

Functions expressed as power series

8.15

Taylor and maclaurin series