Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson 24: The geometric series

Lesson Date:Tuesday, May 12th.

Topic: This lesson covers Chapter 24: The geometric series

WeBWorK: There are two WeBWorK assignments on today’s material, due in one week:Sequences – Geometric

Series – Geometric

**Question of the day:** Can we add up infinitely many numbers?

#### Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## Finite geometric series

Today we look at a new kind of sequence, called a *geometric sequence*, and the corresponding series, *geometric series*.

A geometric sequence is a sequence for which we multiply by a constant number to get from one term to the next, for example:

**Definition 24.1**. A sequence is called a geometric sequence, if any two consecutive terms have a common ratio . The geometric sequence is determined by and the first value . This can be written recursively as:

Alternatively, we have the general formula for the th term of the geometric sequence:

**Example 24.2**. Determine if the sequence is a geometric or arithmetic sequence, or neither or both. If it is a geometric or arithmetic sequence, then find the general formula.

a)

b)

c)

d)

e)

f)

g)

h)

*VIDEO: Introduction to geometric sequences, Example*

**Example 24.3**. Find the general formula of a geometric sequence with the given property

a) and

b) and

c) and is positive

*VIDEO: Finding the formula of a geometric sequence – Example 24.3*

**Example 24.4**. Consider the geometric sequence that is the sequence:

Find the sum of the first 6 terms of this sequence

*VIDEO: Sum of a geometric series – intro example*

**Observation 24.5**. Let be a geometric sequence, whose th term is given by the formula We furthermore assume that Then, the sum is given by

**Example 24.6**. Find the value of the geometric series.

a) Find the sum for the geometric sequence

b) Determine the value of the geometric series:

c) Find the sum of the first 12 terms of the geometric sequence

*VIDEO: The sum of a finite geometric series, Example 24.6*

## Infinite geometric series

Sometimes it makes sense to add up not just a finite number of terms in a sequence, but ALL the terms (infinitely many!).

**Example 24.7**. Consider the geometric sequence

What is the initial term? What is the common ratio?

Let’s try adding up some of the terms. Try this by hand, and by using the formula for finite geometric series. What happens if we add up ALL the terms?

*VIDEO: Infinite geometric series – intro example and formula*

**Definition 24.8**. An infinite series is given by the formula

**Observation 24.9**. Let be a geometric sequence with Then the infinite geometric series is defined whenever . In this case, we have:

**Example 24.10**. Find the value of the infinite geometric series.

a) for

b)

c)

d)

**Example 24.11**. Consider the real number given by . Rewrite this number as an infinite geometric series. Can you figure out what fraction it is equal to?

*VIDEO: Infinite geometric series – examples*

For the geometric sequence homework, when i put in an answer for the nth term, i get “sage failed to evaluate your answer, please notify your instructor.” It also doesn’t allow me to enter the answers in scientific notation

Hi Brandon,

You are right – that’s a WeBWorK error, I believe (your answer is correct). I’ll pass it along to our developer.

Best,

Prof. Reitz

Hi again – our developer suggests it’s probably an issue with traffic on the Sage servers (used to evaluate your answer). As a short-term fix, please try re-submitting your answer every now and then – eventually you will hit a moment when traffic to the Sage server is low, and your answer will be evaluated correctly. Let me know how it goes.

Thanks – sorry for the technical challenges!

-Prof. Reitz

Hey Prof. Reitz

Did you get my exam worksheet, I just sent it is everything ok with it.

Dear Lauwens,

I received your pdf – thanks!

Prof. Reitz