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Get Started Now- Intro Lesson3:26
- Lesson: 1a4:54
- Lesson: 1b13:00
- Lesson: 1c8:19
- Lesson: 1d7:14
- Lesson: 2a4:45
- Lesson: 2b4:47
- Lesson: 2c7:06

In this section, we will take a look at the convergence and divergence of geometric series. We've learned about geometric sequences in high school, but in this lesson we will formally introduce it as a series and determine if the series is divergent or convergent. For the first few questions we will determine the convergence of the series, and then find the sum. For the last few questions, we will determine the divergence of the geometric series, and show that the sum of the series is infinity.

Basic Concepts: Introduction to infinite series, Convergence and divergence of normal infinite series

Related Concepts: Arithmetic series

Formulas for Geometric Series:

$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}$ if -1 < $r$ < 1

$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}$ if -1 < $r$ < 1

If -1 < $r$ < 1, then the geometric series converges. Otherwise, the series diverges.

$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}$ if -1 < $r$ < 1

$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}$ if -1 < $r$ < 1

If -1 < $r$ < 1, then the geometric series converges. Otherwise, the series diverges.

- IntroductionGeometric Series Overview:
- 1.
**Convergence of Geometric Series**

Show that the following series are convergent and find its sum:a)$\sum_{n=0}^{\infty} \frac{1}{3^n}$b)$\sum_{n=1}^{\infty} [(-\frac{5}{8})^{n-1}+(\frac{1+3^n}{7^n})]$c)$\sum_{n=0}^{\infty}4^{n+2}2^{3-4n}$d)$\sum_{n=0}^{\infty} \frac{4^{2(n+2)}}{5^{3n-1}}$ - 2.
**Divergence of Geometric Series**

Show that the following series are divergent:a)$\sum_{n=0}^{\infty} \frac{3^{n-1}}{2^n}$b)$\sum_{n=0}^{\infty}3^{n+2}2^{3-n}$c)$\sum_{n=0}^{\infty}[(\frac{1}{4})^n+(\frac{3}{2})^n2^n]$

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 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