Department of Mathematics : The University of Akron



Lesson Summary:

Students will discover how to find partial sums of an infinite geometric series, as well as the sum of a convergent infinite geometric series through inquiry activities. Students will be able to determine whether certain series are convergent or divergent. They will also be able to convert repeating decimals to fractions using the sum formula for a convergent infinite geometric series.

Key Words:

Partial sums, convergent, divergent, infinite series, geometric series, sum of a convergent infinite geometric series

NCTM Standards Addressed:

Standards: Patterns, Functions, and Algebra Grade 12, Indicators 1 & 2 and Grade 8, Indicator 2

12:1 -- “Analyze the behavior of arithmetic and geometric sequences and series as the number

of terms increases.”

12:2 -- “Translate between the numeric and symbolic form of a sequence or series.”

8:2 – “Generalize patterns and sequences by describing how to find the nth term.”

Benchmarks: Patterns, Functions, and Algebra Grade 8, Benchmark A which state “Generalize and

explain patterns and sequences in order to find the next term and the nth term.” and Patterns,

Functions, and Algebra Grade 12, Benchmark A which states “Analyze functions by investigating

rates of change, intercepts, zeros, asymptotes, and local and global behavior.”

Process Standards: While all the process standards are addressed to some extent, the ones emphasized

in this lesson are Communication, Representation, and Reasoning & Proof.

Learning Objectives: Students will be able to

1) Find partial sums of an infinite geometric series,

2) Find the sum of an infinite geometric series,

3) Determine whether an infinite series, particularly a geometric infinite series, is convergent or divergent,

4) Apply the sum formula for an infinite geometric series to different problem situations, including repeating decimals and word problems.

Prior Knowledge: Students will be required to

1) Know how to identify arithmetic and geometric series,

2) Find the nth term and sum of a finite geometric series,

3) Recognize finite and infinite series,

4) Know how to use the table and function features on a graphing calculator.

Necessary Materials: Ruler with centimeters, graphing calculator

Plan: Day 1 (Part 1 for Block Scheduling)

• Get students into groups, and have them begin on the inquiry activity (Part 1)

• General Time Frame for Worksheet (all approximate):

o Debrief yesterday’s lesson/homework: 10 minutes

o #1 – 7: 7 minutes

o #8 – 10: 4 minutes

o #11 – 14 & Bonus: 8 minutes

o #15 – 17 & Properties, etc.: 7 minutes

o #18 & 19: 14 minutes or finish at home.

• Hints or suggestions:

o For #1 – 7 (especially #2): Be sure the groups are noticing the pattern and working through the “construction” properly.

o For #8 & 15: Be sure the groups are completing the “Write the Partial Series Here” column correctly – it should be an addition problem

o For #12: Groups might have trouble using their function and table features on their calculators

o Be sure to debrief students on properties and definitions, have them write new definitions in their notebooks

Day 2 (Part 2 for Block Scheduling)

• Get students into groups, and have them begin on the inquiry activity (Part 1 & 2)

• General Time Frame for Activity (times are all approximate):

o Debrief Part 1, cover Homework: 10 minutes

o Begin Part 2 Activity, #1 – 4: 12 minutes

o #5(a) & (b): 6 minutes

o Extensions: 22 minutes

• Hints or suggestions:

o For #1 – 5, be sure they get the correct series and are using the formula correctly

o For Extension 1, an alternate proof for this is: if [pic], then according to the Density Property of Real Numbers there should be a number between them. But there is not. So, [pic].

o For Extension 2(d), remember that the initial drop is included in total distance covered as well as two of each distance after each bounce. This is the case because the ball bounces up and covers the same distance as it goes back down.

Assessment:

• Group Participation

• Individual completion of Inquiry Based Activities

• Homework from the textbook with complete answers for all problems and correct answers for a few selected problems

• Questions on the next quiz, with similar difficulty level as the ones included in the activity worksheets

A B

1. What is the length of the entire segment AB? (It will be a whole number of centimeters long)

2. (a) Find the midpoint of [pic], label it M. Write the length of [pic] in the space above it.

(b) Find the midpoint of [pic], label it N. Write the length of [pic] in the space above it.

(c) Find the midpoint of [pic], label it P. Write the length of [pic] in the space above it.

(d) Find the midpoint of [pic], label it Q. Write the length of [pic] in the space above it.

(e) Find the midpoint of [pic], label it R. Write the length of [pic] in the space above it.

(f) Find the midpoint of [pic], label it S. Write the length of [pic] in the space above it.

3. Write the lengths found in number 2 as a sequence.

4. Predict the next 3 terms of this sequence.

5. Is the sequence arithmetic, geometric, or neither? Justify your conclusion.

6. Write a rule for the nth term of the sequence.

7. Ignoring physical limitations, could this sequence continue infinitely? In other words, is this the beginning of an infinite sequence? Explain your answer.

Remember a series is the sum of the terms of a sequence. A partial sum, Sn, of an infinite series is the sum of its first n terms.

Now, let’s treat the sequence developed in #1 – 7 as a series.

8. Find the following partial sums of our sequence.

|n |Write the Partial Series Here |Sn |

| | | |

|1 | | |

| | | |

|2 | | |

| | | |

|3 | | |

| | | |

|4 | | |

| | | |

|5 | | |

| | | |

|6 | | |

| | | |

|7 | | |

| | | |

|8 | | |

| | | |

|9 | | |

9. What do you notice about the partial sums as more and more terms are added? And will the partial sums increase without limit?

10. Using the formula for the sum of a geometric series, show that the sum of the first 20 terms is approximately 15.9999847. Explain why this supports or fails to support your conclusion from #9.

This geometric series, although infinite, has a finite sum. Consider the following questions to determine how to calculate that sum.

11. What is r, the common ratio, for the series?

12. As n ( [pic], what does rn approach for this series? Use the table feature on your graphing calculator to support your conclusion. On the TI – nspire, write a function in the “Graphs & Geometry” page for rn. Then from the “Menu”, choose “2:View”, then “8:Add Function Table”. From here, select “Menu”, then “3:Edit Function Table Settings”, then change the Independent variable to “Ask”. Another way is to insert values for n into the function from the “Calculator” screen – just type “f1(x – value)” and enter.

13. As n ( [pic], the sum formula, [pic] will change. Write and simplify the new formula, based on the results of #12. This will be the sum formula for an infinite geometric series.

14. Using this new formula, determine the sum of all the terms in this infinite series.

Maybe you find it troubling that an infinite series has a finite sum. If you do, then you are not alone. An ancient Greek mathematician, Zeno of Elea, was also troubled by such apparent contradictions. So much so that he is famous for his eponymous paradoxes, Zeno’s Paradoxes. To find out more about Zeno, visit the website www-history.mcs.st-andrews.ac.uk/Biographies/Zeno_of_Elea.html

Think back to the original line segment in this worksheet. Does it not have a definite length? Yet you can divide it in half infinitely many times, but that does not change its length. And all those “half” line segments add up to the total length.

BONUS: What theorem from geometry supports the idea that all the infinite amount of segments will combine to make the total segment length?

Let’s now consider the infinite series,

[pic]

15. Find the 1st 5 partial sums of this series and record in the table below.

|n |Write the Partial Series Here |Sn |

|1 | | |

|2 | | |

|3 | | |

|4 | | |

|5 | | |

16. What do you notice about the partial sums as more and more terms added?

17. Will this series have a finite sum? Justify your conclusion by looking at the rule for this series, as well as the table of partial sums.

Property. An infinite geometric series will have a sum, if |r| < 1. And it will not have a sum if |r| > 1.

Definitions. A convergent infinite series is an infinite series with a finite sum. A divergent infinite series is an infinite series that does not have a finite sum.

Formula for the Sum of an Infinite Geometric Series. Just in case you struggled with #13 and 14, here is the formula for the sum of a convergent infinite geometric series.

[pic]

Notice two things about this formula. (1) It is easier to calculate the sum of an infinite amount of terms than a finite number of terms; (2) there should be no subscript on S because there is an infinite number of terms.

18. Practice Problems. Find the sum, if it exists, for the following series.

a) The first term of an infinite geometric sequence is 3, and the common ratio is [pic].

b) [pic]

c) [pic]

d) [pic]

19. More Practice. Use what you know about a geometric series, and the sum formula of an infinite geometric series to find the following information regarding different infinite geometric series.

a) The sum of an infinite geometric series is 81 and its common ratio is [pic]. Find the first three terms of the series.

b) The sum of an infinite geometric series is 125 and its 7th and 8th terms are [pic] and [pic], respectively. Find the first three terms of the series

c) The first term of an infinite geometric series is – 8, and its sum is [pic]. Find the first four terms of the series.

An infinite geometric series can be used to model a repeating decimal. For example,

[pic]

1. Write the series above in summation notation. Identify [pic]and [pic].

2. Find the sum of the series referred to by # 1, as a fraction in lowest terms. The result will be the repeating decimal’s fraction equivalent. Check your answer and explain how you checked your answer.

3. Try this procedure with the well known repeating decimal [pic] to find its fraction equivalent.

4. Now use this procedure to convert the less familiar decimal [pic].

Notice [pic] is always the first set of digits that repeat. And [pic]is always 10 to the negative amount of repeating places. For example, in the repeating decimal[pic], [pic]and [pic] -- since five places repeat.

5. Convert [pic] to a fraction.

Consider the next example, in which not every place of the decimal repeats. Convert [pic] to a fraction. First consider the series that models this decimal, it is

[pic][pic]. 0.5 is not part of the repeating pattern so it is not part of the infinite geometric series used to covert that decimal. The first term of that series is 0.043, and since 2 places repeat, r = 0.01.

[pic] [pic]

6. Try these examples. Convert each of these decimals to a fraction.

(a) [pic] (b) [pic]

Extension. Using the infinite geometric sum formula, prove [pic].

Extension. A ball is dropped from a height of 10 feet. Each time it strikes the ground, it bounces up to 80% of the previous height.

a) What height will the ball bounce to after it strikes the ground for the third time?

b) How high will it bounce after it strikes the ground for the nth time?

c) How many times does the ball need to strike the ground before its bounce is less than one inch?

d) What total distance does the ball travel before it stops bouncing (This may be more difficult than it appears, so be careful)?

Extension. Consider the series 8 + 4 + 2 + 1 + … developed in Part 1 of this activity. Recall that the sum was determined to be 16. Now consider the sequence of partial sums for this series: 8, 12, 14, 15, 15.5, 15.75, …

a) Write a rule for the sequence of partial sums.

b) Graph the series on your graphing calculator on the window [0,20] X [0,17].

Your graph should look like the one shown here from the TI – nspire.

[pic]

c) Does there appear to be an asymptote for this series? If so, what is it?

d) Considering the result of part (c), explain why an infinite geometric series has a finite sum, not approaches a sum, which is the apparent conclusion from part (c). Remember what series you have just graphed and analyzed. Try to summarize the results in everyday language.

Review. Identify and write all properties, definitions, and formulas learned in this activity (both parts 1 & 2).

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Lesson Plan for Infinite Geometric Series

Andrew Stephens and Janice Stebner

The Infinite Geometric Series PART 1

The Infinite Geometric Series PART 2

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