Worksheet: Infinite geometric series
Mr. Chvatal Name:__________________________
The sum of an infinite geometric series is the sum of the terms of an infinite geometric sequence, and is defined as:
S = a + ar + ar
2
+ ...
+ ar n
+ ...
where a is the first term, r is the ratio of successive terms and S is the sum of the terms.
The formula for the sum of the terms of an infinite geometric series is:
S a
=
1 − r
, where r ≠ 1 , or in sigma notation as
∞
ar i i = 0
Example: Calculate the sum of the sequence -2, 1, -1/2, 1/4, -1/8, 1/16, ...
S
− 2
=
1
 1 
− −
 2 
= −
4
3
Note that we can also write the sum as
∞
( − 2 )
 1  i

−
2  i = 0
Present value
You can calculate the present value P of a future payment B using infinite geometric series.
Example: How much money would you need to deposit in the bank today to cover annual payments of $3 million forever, given an interest rate of 5%?
Use the formula P =
B



1 −


1
1 + r





, where r is the interest rate.
3
P =

1 −


1 



= $63 million

Example: How much money would you need to deposit in the bank today to cover just 20 annual payments of $3 million, given an interest rate of 5%?
Use the formula P =
B

 1 −

 1
1
+ r
 n 

, where r is the interest rate and n is

1 −


1
1 + r



 the number of payments.
P =

3 1 −





1 −


1
1


20







≈ $39.255
million
4)
3)
Problems
Calculate the sum of the following sequences.
9, 3, 1, 1/3, 1/9,… 1)
2) -5, 1, -1/5, 1/25,…
What is the present value of a $200,000 annual payment to be paid forever, given an interest rate of 4%?
What is the present value of a $5,000 annual payment to be paid for 30 years, given an interest rate of 3.2%?