12.5. Geometric Series http://www.ck12.org
0 + 0 + 0 + 0 +···
Example C
What is the sum of the first 8 terms in the following geometric series?
4 + 2 + 1 +^12 +···
Solution:The first term is 4 and the common ratio is^12.
SU M=a 1 (^11 −−rrn)= 4
( 1 −( 1
1 −^21 )^8
2)
= 4
( (^255256)
(^12)
)
=^25532
Concept Problem Revisited
An infinite geometric series converges if and only if|r|<1. Infinite arithmetic series never converge.
Vocabulary
Toconvergemeans the sum approaches a specific number.
Todivergemeans the sum does not converge, and so usually goes to positive or negative infinity. It could also mean
that the series oscillates infinitely.
Apartial sumof an infinite sum is the sum of all the terms up to a certain point. Considering partial sums can be
useful when analyzing infinite sums.
Guided Practice
- Compute the sum from Example A using the infinite summation formula and confirm that the sum truly does
converge. - Does the following geometric series converge or diverge? Does the sum go to positive or negative infinity?
− 2 + 2 − 2 + 2 − 2 +··· - You put $100 in a bank account at the end of every year for 10 years. The account earns 6% interest. How much
do you have total at the end of 10 years?
Answers: - The first term of the sequence isa 1 = 0 .2. The common ratio is 0.1. Since| 0. 1 |<1, the series does converge.
- 2 ( 1 −^10. 1 )=^00 ..^29 =^29
- The initial term is -2 and the common ratio is -1. Since the|− 1 |≥1, the series is said to diverge. Even though
the series diverges, it does not approach negative or positive infinity. When you look at the partial sums (the sums
up to certain points) they alternate between two values:
− 2 , 0 ,− 2 , 0 ,...
This pattern does not go to a specific number. Just like a sine or cosine wave, it does not have a limit as it approaches
infinity. - The first deposit gains 9 years of interest: 100· 1. 069
The second deposit gains 8 years of interest: 100· 1. 068. This pattern continues, creating a geometric series. The
last term receives no interest at all.
100 · 1. 069 + 100 · 1. 068 +··· 100 · 1. 06 + 100
Note that normally geometric series are written in the opposite order so you can identify the starting term and the
common ratio more easily.