UNIT 1 - ARITHMETIC & GEOMETRIC SEQUENCES
Task #10 – Derive Formula for Infinite Geometric Series
Common Core: HS.A-SSE.B.4, HS.F-BF.A.2
MA40: ALGEBRA 2
Name:
Period:
INVESTIGATION
Convergence – An infinite series is convergent if the sequence of its partial sums S1, S2 , S3 , ...
approaches a given number.
Divergence – An infinite series is divergent if the sequence of its partial sums S1, S2 , S3 , ... do not
approach a given number.
I) Investigate the series represented by the summation notation by completing the table.
Use your calculator to complete the table.
n
1 i1
3

i 1 2
This is a(n) _______________ series. We will use the formula ____________________
to find the sum of the first n terms of the series.
S n is the ________________________________________. n is the __________________________.
a1 is the _______________ & equals ___________. r is the ________________ & equals _________.
n
Use x on
calculator.
Input value.
rn
Use Y1 on
calculator
2nd Y 
Sn
n
Expand the series
Y2 on calculator
2nd Y  screen.
Use
1
 2 3
i 1
.
i 1
Consider the ending terms of the series.
screen.
n 1
n2
n3
1
2
1 3
S2  
2 2
1 3 9
S3   
2 2 2
S1 
1 3 9
   ... 
2 2 2
1 3 9
S9     ... 

n9
2 2 2
1 3 9
S10     ... 


n  10
2 2 2
Question #1 – Consider column 4. What do you notice about the last term of each series as n
increases? Do you think this trend will continue?
n 8
S8 
Question #2 – Consider column 3. As the number of terms n increases, what do you notice about the
sum S n .
Task #10 – Derive Formula for Infinite Geometric Series – (continued)
Question #3 – Will the values in column 3 continue to grow without bound or will they eventually “equal”
some value? Does the series represented by the summation notation represent a convergent or
divergent series? Explain.
II) Investigate the series represented by the summation notation by completing the table.
Use your calculator to complete the table.
1
3 

2
i 1
n
i 1
This is a(n) ______________ series. We will use the formula ___________________
to find the sum of the first n terms of the series.
S n is the ________________________________________. n is the __________________________.
a1 is the _______________ & equals ___________. r is the ________________ & equals _________.
n
Use x on
calculator.
Input value.
rn
Use Y1 on
calculator
2nd Y 
Sn
1
3 

2
i 1
n
Expand the series
Y2 on calculator
2nd Y  screen.
Use
i 1
.
Consider the ending terms of the series.
screen.
n 1
S3  3
n2
S2  3 
n3
3
2
3 3
S3  3  
2 4
3 3
  ... 
2 4
3 3
S12  3    ... 

n  12
2 4
3 3
S13  3    ... 


n  13
2 4
Question #4 – Consider column 4. What do you notice about the last term of each series as n
increases? Do you think this trend will continue?
n  11
S11  3 
Question #5 – Consider column 3. As the number of terms n increases, what do you notice about the
sum S n .
Question #6 – Will the values in column 3 continue to grow without bound or will they eventually “equal”
some value? Does the series represented by the summation notation represent a convergent or
divergent series? Explain.
Task #10 – Derive Formula for Infinite Geometric Series – (continued)
Question #7 – Consider column 2. What do you notice about the value of r n as the number of terms n
increases? Do you think this has anything to do with your answer to Question #6? Explain.
DEVELOPING THE MATH CONCEPTS & TERMS
Deriving the Formula for the Sum of an Infinite Geometric Series.
Finding the Sums of Infinite Geometric Series

1) Find
 3  0.7 
i 1
i 1
1 1 1
2) Find the sum of 1     ...
4 16 64
Finding the Common Ratio
3) An infinite geometric series with first term a1  4 has the sum of 10.
What is the common ratio of the series?
Task #10 – Derive Formula for Infinite Geometric Series – (continued)
Using an infinite Series as a Model
3) A ball is dropped from a height of 10 feet. Each time it hits the ground,
it bounces to 80% of its previous height. Find the total distance
traveled by the ball.
4) Writing a Repeating Decimal as a Fraction
Write 0.181818... as a fraction
Summary: