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- Sequence and Series

Still Confused?

Try reviewing these fundamentals first.

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Try reviewing these fundamentals first.

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Get Started Now- Lesson: 1a10:22
- Lesson: 1b11:59
- Lesson: 2a11:52
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- Lesson: 3a13:41
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- Lesson: 4a23:18
- Lesson: 4b7:39

In this lesson, we will learn about what a power series is. Power series have coefficients, x values, and have to be centred at a certain value a. Our goal in this section is find the radius of convergence of these power series by using the ratio test. We will call the radius of convergence L. Since we are talking about convergence, we want to set L to be less than 1. Then by formatting the inequality to the one below, we will be able to find the radius of convergence. Lastly, we will learn about the interval of convergence. The interval of convergence is the value of all x's for which the power series converge. Also make sure to check the endpoint of the interval because there is a possibility for them to converge as well.

Note *Power Series are in the form:

$\sum_{n=0}^{\infty}c_n(x-a)^n$

where $c_n$ are the coefficients of each term in the series and $a$ is number.

To find the**Radius of Convergence** of a power series, we need to use the ratio test or the root test. Let $A_n=c_n(x-a)^n$. Then recall that the ratio test is:

$L=\lim$_{n →$\infty$}$|\frac{A_{n+1}}{A_n}|$

and the root test is

$L=\lim$_{n →$\infty$}$|A_n|^{\frac{1}{n}}$

where the**convergence** happens at $L$< $1$ for both tests. More accurately we can say that the **convergence** happens when $|x-a|$ < $R$, where is the **Radius of Convergence**.

The**Interval of Convergence** is the value of all $x$’s, for which the power series converges. So it is important to also check if the power series converges as well at $|x-a|=R$.

$\sum_{n=0}^{\infty}c_n(x-a)^n$

where $c_n$ are the coefficients of each term in the series and $a$ is number.

To find the

$L=\lim$

and the root test is

$L=\lim$

where the

The

- 1.Radius and Interval of Convergence with Power Series Overviewa)Radius and Interval of Convergenceb)Checking the Endpoints for the Interval of Convergence
- 2.
**Questions based on Radius of Convergence**

Determine the radius of convergence for the following power series:a)$\sum_{n=0}^{\infty}\frac{2^nx^n}{n!}$b)$\sum_{n=0}^{\infty}3^n|x+3|^{2n+1}$ - 3.
**Radius of Convergence of Sine and Cosine**

Determine the radius of convergence for the following power series:a)$\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$b)$\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$ - 4.
**Questions based on Interval of Convergence**

Determine the interval of convergence for the foll owing power series:a)$\sum_{n=0}^{\infty}\frac{n+2}{3^n}(x+5)^n$b)$\sum_{n=0}^{\infty}\frac{(2x-3)^n}{n^n}$

5.

Sequence and Series

5.1

Introduction to sequences

5.2

Monotonic and bounded sequences

5.3

Introduction to infinite series

5.4

Convergence and divergence of normal infinite series

5.5

Convergence & divergence of geometric series

5.6

Convergence & divergence of telescoping series

5.7

Divergence of harmonic series

5.8

P Series

5.9

Alternating series test

5.10

Divergence test

5.11

Comparison & limit comparison test

5.12

Integral test

5.13

Ratio test

5.14

Root test

5.15

Absolute & conditional convergence

5.16

Radius and interval of convergence with power series

5.17

Functions expressed as power series

5.18

Taylor series and Maclaurin series

5.19

Approximating functions with Taylor polynomials and error bounds

We have over 170 practice questions in Calculus 2 for you to master.

Get Started Now5.1

Introduction to sequences

5.2

Monotonic and bounded sequences

5.4

Convergence and divergence of normal infinite series

5.5

Convergence & divergence of geometric series

5.6

Convergence & divergence of telescoping series

5.7

Divergence of harmonic series

5.8

P Series

5.9

Alternating series test

5.10

Divergence test

5.11

Comparison & limit comparison test

5.12

Integral test

5.13

Ratio test

5.14

Root test

5.15

Absolute & conditional convergence

5.16

Radius and interval of convergence with power series

5.17

Functions expressed as power series

5.18

Taylor series and Maclaurin series