FOCUS: Determine the sum of an infinite geometric series.
An infinite geometric series has an infinite number of terms.
For an infinite geometric series, if the sequence of partial sums converges to a constant value as the number of terms increases, then the geometric series is convergent and the constant Value is the finite sum of the series. This sum is called the sum to infinity and is denoted by S∞.
Example 1: Estimating the Sum of an Infinite Geometric Series
Determine whether each infinite geometric series has a finite sum. Estimate each finite sum.
a) ½+¼+⅛+1/16+⋯
b) 0.5+1+2+4+⋯
c) ½-¼+⅛-1/16+⋯
solution:
For each geometric series, calculate some partial sums.
The next term in the series is 1/32.
These partial sums appear to get closer to 1.
An estimate of the finite sum is 1.
As the number of terms increases, the partial sums increase, so the series does not have a finite sum.
The partial sums alternately increase and decrease, but appear to get close to 0.33… .
An estimate of the finite sum is 0.33… .
In Example 1, each series has the same first term but different common ratios. It appears that the Value of r determines whether an infinite geometric series converges or diverges.
Consider the rule for the sum of n terms.
For a geometric series,
When -1<r<1, rn approaches 0 as n increases indefinitely.
The Sum of an Infinite Geometric Series
For an infinite geometric series with first term, t1, and common ratio, -1<r<1, the sum of the series, S∞, is:
Ex2 Find the sum to infinity for the sequence tn: {10, 1, 0.1, …}.
Solution:
Ex2 (1) Write the formula for the nth term of the geometric sequence.
(2) From the question we know that the first term, a, is 10 and r=0.1.
(3) Write the formula for the sum to infinity.
(4) Substitute a=10 and r=0.10 into the formula and evaluate.
Ex3: Determining the Sum of an Infinite Geometric Series
Determine whether each infinite geometric series converges or diverges. If it converges, determine its sum.
a) 27-9+3-1+…
b) 4-8+16-32+…
solution:
a) 27-9+3-1+…
The common ratio is between -1 and 1, so the series converges. Use the rule for the sum of an infinite geometric series:
The sum of the infinite geometric series is 20.25.
b) 4-8+16-32+…
The common ratio is not between -1 and 1, so the series diverges. The infinite geometric series does not have a finite sum.
Infinite Geometric Series
When a series has an infinite number of terms, it is called an infinite series and the sum of the series is called the sum to infinity of the series.
Let us consider the value of a proper fraction (less than 1) if we keep multiplying it by itself. Take for example, ¼, and keep multiplying it by itself, i.e. ¼n, as n increases indefinitely. We can represent this situation in a table using a calculator.
From the table we can see that the bigger the value of n, the nearer ¼n gets to 0.
(This will happen for any proper fraction, positive or negative.)
We say that the limit of ¼n, as n approaches infinity, is O.
Symbolically:
n→∞ means ‘as n approaches infinity’.
lim is short for limit.
In general, for the infinite geometric series:
if r is a proper fraction, then the terms will get closer to zero.
For r to be a proper fraction it must be between-1 and 1, i.e., -1<r<1. ∴ If -1<r<1 then
Notes: If r>1 or r<-1, then limn→∞rn does not exist.
The sum to infinity, S∞, of a series is denoted by limn→∞Sn.
If limn→∞Sn exists, the series is said to be convergent.
If limn→∞Sn does not exist, the series is said to be divergent.
Let us now develop the general formula for the sum to infinity of a geometric series in which -1<r<1.
The only part of this formula that changes as n increases is rn.
As, n→∞, rn→∞, because r is a proper fraction.
Sum to infinity of a geometric series
Note: -1<r<1 is often written |r|<1. Infinite Geometric Sequence:
A geometric sequence in which the number of terms are infinite is called as infinite geometric sequence. For example:
Infinite Series:
Consider a geometric sequence a, a⋅r, ar2, … to n terms. Let Sn denote the sum of n terms then Sn=a+a⋅r+a⋅r2+… to n terms.
Formula
Taking limit as n→∞ on both sides
as |r|<1, n→∞, rn→0
Therefore the formula for the sum of infinite terms of geometric sequence is given by:
Convergent Series:
An infinite series is said to be the convergent series when its sum tends to a finite and definite limit.
Divergent Series:
When the sum of an infinite series is infinite, it is said to be the Divergent series.
The limiting sum of a geometric series
We have seen that the sum of the first n terms of a geometric series with first term a and common ratio r is
In the case when r has magnitude less than 1, the term rn approaches 0 as n becomes Very large. So, in this case, the sequence of partial sums S1, S2, S3, ⋯ has a limit:
The Value of this limit is called the limiting sum of the infinite geometric series. The Values of the partial sums Sn of the series get as close as we like to the limiting sum, provided n is large enough.
The limiting sum is usually referred to as the sum to infinity of the series and denoted by S∞. Thus, for a geometric series with common ratio r such that |r|<1, we have
Ex4. (Fans) A fan is running at 10 revolutions per second. After it is turned off, its speed decreases at a rate of 75% per second. Determine the number of revolutions completed by the fan after it is turned off.
Solution:
Given a1=10 and r=100%-75% or 0.25.
Find the sum.
The fan completed 40/3 revolutions after it is turned off.
Answer: 40/3
Ex5. (Rechargeable Batteries)
A certain rechargeable battery is advertised to recharge back to 99.9% of its previous capacity with every charge. If its initial capacity is 8 hours of life, how many total hours should the battery last?
Solution:
Given a1=8 and r=99.9% or 0.999
Find the sum.
Answer: 8000 hrs
Ex6. (Multiple Representations)
In this problem, you will use a square of paper that is at least 8 inches on a side.
a. (Concrete) Let the square be one unit. Cut away one half of the square. Call this piece Term 1. Next, cut away one half of the remaining sheet of paper. Call this piece Term 2. Continue cutting the remaining paper in half and labeling the pieces with a term number as long as possible. List the fractions represented by the pieces.
b. (Numerical) If you could cut the squares indefinitely, you would have an infinite series. Find the sum of the series.
c. (Verbal) How does the sum of the series relate to the original square of paper? Solution:
a. ½, ¼, ⅛, 1/16, …
b. Find r.
c. The original square has area 1 unit and the area of all the pieces cannot exceed 1.
Answer:
a. ½, ¼, ⅛, 1/16, …
b. 1
c. The original square has area 1 unit and the area of all the pieces cannot exceed 1.
Ex7. (Physics) In a physics experiment, a steel ball on a flat track is accelerated, and then allowed to roll freely. After the first minute, the ball has rolled 120 feet. Each minute the ball travels only 40% as far as it did during the preceding minute. How far does the ball travel?
Solution:
Given a1=120 and r=40% or 0.4.
Find the sum.
The ball travels 200 ft.
Answer: 200 ft
Ex8. (Cars) During a maintenance inspection, a tire is removed from a car and spun on a diagnostic machine. When the machine is turned off, the spinning tire completes 20 revolutions the first second and 98% of the revolutions each additional second. How many revolutions does the tire complete before it stops spinning?
Solution:
Given a1=20 and r=98% or 0.98
Find the sum.
The tire completes 1000 revolutions.
Answer: 1000 revolutions
Ex9. (Economics) A state government decides to stimulate its economy by giving $500 to every adult. The government assumes that every one who receives the money will spend 80% on consumer goods and that the producers of these goods will in turn spend 80% on consumer goods. How much money is generated for the economy for every $500 that the government provides?
Solution:
Here, a1=500 and r=80% or 0.8.
Find the sum.
Answer: $2500
Ex10. (Science Museum) An exhibit at a science museum offers visitors the opportunity to experiment with the motion of an object on a spring. One visitor pulls the object down and lets it go. The object travels 1.2 feet upward before heading back the other way. Each time the object changes direction, it decreases its distance by 20% when compared to the previous direction. Find the total distance traveled by the object.
Solution:
Here a1=1.2, r=1-0.2 or 0.8.
The total distance traveled by the object is 6 feet.
Answer: 6 ft
Ex11: Finding the Sum of an Infinite Geometric Series
Find the sum of the infinite geometric series:
Solution:
Before finding the sum, we must find the common ratio.
Because r=-½, the condition that |r|<1 is met.Thus, the infinite geometric series has a sum. This is the formula for the sum of an infinite geometric series. Let a1=⅜ and r=-½.
Thus, the sum of ⅜-3/16+3/32-3/64+⋯ is ¼. Put in an informal way, as we continue to add more and more terms, the sum is approximately ¼.
Ex12. What is the sum of an infinite geometric series with a first term of 27 and a common ratio of ⅔?
A 18
B 34
C 41
D 65
E 81
Solution:
Given a1=27, r=⅔.
Find the sum.
Option E is the correct answer.
Answer: E
Ex13: Sum of an lnfinife Geometric Series
Find the sum of each infinite geometric series, if it exists.
a. ½+⅜+9/32+⋯
First, find the value of r to determine if the sum exists.
a1=½ and a2=⅜, so r=⅜⁄½ or ¾. Since |¾|<1, the sum exists.
Now use the formula for the sum of an infinite geometric series.
Sum formula
The sum of the series is 2.
b. 1-2+4-8+…
a1=1 and a2=-2, so r=(-2)/1 or -2. Since |-2|≥1, the sum does not exist.
Key Point
The sum to infinity of a geometric sequence with starting value a and common ratio r is given by
where -1<r<1.
Ex14. Find the sum to infinity of the geometric sequence
Solution:
For this geometric sequence we have a=1 and r=⅓. As -1<r<1 we can use the formula, so that
Ex15. Find the sum of infinite geometric series in which a=128, r=-½
Solution:
Sum of the infinite number of a geometric series
If -1<r<1 and the infinite geometric series a+ar+ar2+⋯, then sum of the infinite geometric series is
Ex16: Find the sum of the infinite geometric sequence
Solution: Here a=1, r=-⅓
Thus, sum of the infinite geometric sequence is
Ex16. Find the limiting sum for the geometric series
a) 1+⅓+1/9+⋯
b) 8-6+9/2-…
Solution
a) Here a=1 and r=⅓, so the limiting sum exists and is equal to
b) Here a=8 and r=-¾, so the limiting sum exists and is equal to
Ex17. Find the sum to infinity in each of the following Geometric Sequences.
Ex17a)
Answer:
Using
Ex17b)
6, 1.2, 0.24, …
Answer:
Let S=6+1.2+0.24+⋯
Here, a=6 and r=0.2.
Using
Ex17c)
Answer:
Here, a=-¾ and r=-¼.
Using
Ex17d)
Prove that: 3½×3¼×3⅛×…=3
Answers:
3(½+¼+⅛+⋯)
Power of 3 is in the form of a geometric series with a=½ and r=½.
½+¼+⅛+⋯
Using
