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Calculus

Introduction to infinite seriesCalculus

Convergence & divergence of geometric series Calculus

Divergence of harmonic seriesCalculus

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- Integral Calculus
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Calculus

Introduction to infinite seriesCalculus

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P SeriesStill Confused?

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Calculus

Introduction to infinite seriesCalculus

Convergence & divergence of geometric series Calculus

Divergence of harmonic seriesCalculus

P SeriesNope, got it.

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Get Started Now- Intro Lesson: a5:34
- Intro Lesson: b2:20
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- Lesson: 1c11:51
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- Lesson: 2b8:06
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You may notice that some series look very complicated, but it shares the same properties as another series that looks very simple and easy. In this case, we can use the comparison test or limit comparison test. We will look at what conditions must be met to use these tests, and then use the tests on some complicated looking series. Lastly, we will use both the comparison test and the limit comparison test on a series, and conclude that they give the same result.

Basic Concepts:Introduction to infinite series, Convergence & divergence of geometric series , Divergence of harmonic series, P Series,

Note *The Comparison test says the following:

Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\leq b_n$ for all $n$ and $a_nb_n\geq0$. Then we say that

1. If $\sum b_n$ is convergent, then $\sum a_n$ is also convergent

2. If $\sum a_n$ is divergent, then $\sum b_n$ is also divergent.

The Limit Comparison Test says the following:

Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\geq 0$ and $b_n$ > 0 for all $n$. Then we say that

$\lim$_{n →$\infty$} $\frac{a_n}{b_n}=c$

If $c$ is a positive finite number, then either both series converge or diverge.

Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\leq b_n$ for all $n$ and $a_nb_n\geq0$. Then we say that

1. If $\sum b_n$ is convergent, then $\sum a_n$ is also convergent

2. If $\sum a_n$ is divergent, then $\sum b_n$ is also divergent.

The Limit Comparison Test says the following:

Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\geq 0$ and $b_n$ > 0 for all $n$. Then we say that

$\lim$

If $c$ is a positive finite number, then either both series converge or diverge.

- IntroductionOverview:a)Comparison testb)Limit Comparison test
- 1.
**Convergence & Divergence of Comparison Tests**

Use the Comparison Test to determine if the series converge or diverge.a)$\sum_{n=1}^{\infty}\frac{1}{2^n+5}$b)$\sum_{n=1}^{\infty}\frac{n^4+5}{n^5-sin^4(2n)}$c)$\sum_{n=1}^{\infty}\frac{n^4cos^4(7n)-1}{n^6}$ - 2.
**Convergence & Divergence of Limit Comparison Tests**

Use the Limit Comparison Test to determine if the series converge or diverge.a)$\sum_{n=3}^{\infty}\frac{n^2+n^3}{\sqrt{n^8+n^4}}$b)$\sum_{n=1}^{\infty}\frac{1}{n^2-7n-12}$ - 3.
**Understanding of Both Tests**

Use both the comparison and limit comparison test for the series $\sum_{k=1}^{\infty}\frac{\sqrt{k^3-1}}{k^3-2k^2+5}$. What do both tests say?

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 Integral Calculus 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