CONDENSED
LE SSO N
9.1
Arithmetic Series
In this lesson you will
●
●
learn the terminology and notation associated with series
discover a formula for the partial sum of an arithmetic series
A series is the indicated sum of terms of a sequence. For example, consider
the sequence
u1 4
un un1 2
where n 2
The sum of the terms in this sequence is the series
u1 u 2 u3 u4 · · ·
4 6 8 10 · · ·
or
The sum of the first n terms in a series is represented by Sn. For example,
S6 u1 u 2 u3 u4 u5 u6 4 6 8 10 12 14 54
The sum of any finite, or limited, number of terms is called a partial sum
6
of the series. The notations S6 and
u1 u 2 u3 u4 u5 u6.
un are shorthand ways of writing
n 1
To find the sum of the integers from 1 to 100, you could add the terms one by
one. You can use technology and a recursive formula to do this quickly. First,
write a recursive definition for the sequence of positive integers.
Sequence:
u1 1
un un1 1
where n 2
Then, write the definition for the related series. Remember, the sum of the first
100 terms is the sum of the first 99 terms plus the 100th term.
Series:
S1 1
Sn Sn1 un where n 2
The table shows each term in the sequence and the sequence of partial sums. The
points on the graph represent the partial sums S1 through S100. You can use either the
table or the graph to find that S100, the sum of the integers from 1 to 100, is 5050.
(continued)
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Lesson 9.1 • Arithmetic Series (continued)
In the investigation you will find a formula for finding a partial sum of an
arithmetic series without finding all the terms and adding.
Investigation: Arithmetic Series Formula
Work through Steps 1 and 2 of the investigation in your book. If you have the
materials, complete the rest of the investigation. Then check your work against
the solution below.
The length of the first step is 4, the second is 7, and so on until the last
step, which is 16.
Step 1
Sequence: 4, 7, 10, 13, 16
Sum of the series: 4 7 10 13 16 50
Step 2 The dimensions of the rectangle are 20 units by 5 units. Note that the
area is 100 square units, twice the value of the sum of the series.
Slide
u1
u2
u3
u4
u5
Use the sequence 2, 4, 6, 8. Then u1 2, d 2. Note that the
related series is 2 4 6 8 20. The figure below shows two copies of a
step-shaped figure representing the sequence. The dimensions of the rectangle are
10 units by 4 units, giving an area of 40 square units.
Steps 3 and 4
Slide
u1
u2
u3
u4
The area of the rectangle is given by n ⭈ u1 u4. The length of the rectangle
is equal to the sum of the first and last terms of the sequence, u1 u4, and the
height of the rectangle is equal to n, the number of terms in the sequence.
n ⭈ u1 un
Step 5 The partial sum, Sn , of an arithmetic series is Sn _________
. This is
2
one-half of the area of the rectangle.
Use the formula from the investigation to verify that the sum of the integers from
1 to 100 is 5050. Then read the example in your book and the text following it.
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CONDENSED
LE SSO N
9.2
Infinite Geometric Series
In this lesson you will
●
●
learn that some infinite geometric series converge to a long-run value,
or sum
discover a formula for finding the sum of a convergent geometric series
In Lesson 9.1, you found partial sums of arithmetic series. If you start adding
terms of an arithmetic sequence, the magnitude of the partial sum increases. This
eventually happens even if the terms are small, as in 0.001, 0.002, 0.003, and so
on. This is not always the case with a geometric series.
A geometric series is the summation of terms in a geometric sequence. For
example, consider the geometric sequence
1 , __
1, ___
1 , ___
1 , ___
1 , ___
1 ,...
1, __
__
2 4 8 16 32 64 128
This series has a constant ratio of 21, so the terms get smaller and smaller. You can
add the terms to create a geometric series. Here are some of the partial sums:
3
1 __
1 __
S2 __
4
4
2
7
1 __
1 __
1 __
S3 __
4
2
8
8
15
1 __
1 ___
1 __
1 ___
S4 __
4
2
8
16
16
31 __
63 ___
127
If you continue to find partial sums, you will get __
32 , 64 , 128 , and so on. Although
the partial sums get larger and larger, they are always less than 1. It appears that
if you add an infinite number of terms, the result will not be infinite.
An infinite geometric series is a geometric series with an infinite number of
terms. A convergent series is a series for which the sequence of partial sums
approaches a long-run value as the number of terms increases. This long-run
1
1
__
value is the sum of the series. The series _12 _14 _18 __
16 32 is a
convergent series with a long-run value, or sum, of 1. Work through Example A
in your book.
Investigation: Infinite Geometric Series Formula
Work through the investigation yourself before reading the solutions below.
0.04
1
__
The first term, u1, is 0.4. The common ratio, r, is ___
0.4 10 , or 0.1. The
multiplier and r are reciprocals. You could use any power of ten as a multiplier.
Step 1
Step 2 Let S 0.4444 . . . . Then 0.1S = 0.0444 . . . . Subtract S and 0.1S and
then solve for S.
S 0.4444 . . .
0.1S 0.0444 . . .
0.9S 0.4
0.4 , or __
4
S ___
0.9
9
4.
This method still resulted in S __
9
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Lesson 9.2 • Infinite Geometric Series (continued)
Step 3 The first term, u1, is 0.9. The ratio, r, is 0.1. Let S 0.9999 . . . and
0.1S 0.0999 . . . . Subtract S and 0.1S and then solve for S.
S 0.9999 . . .
0.1S 0.0999 . . .
0.9S 0.9
S1
Step 4 The first term, u1, is 0.27. The ratio, r, is 0.01. Let S 0.272727 . . . and
0.01S 0.002727 . . . . Subtract and then solve for S.
S 0.272727 . . .
0.01S 0.002727 . . .
0.99S 0.27
0.27 ___
27 ___
3
S ____
0.99
99
11
If S u1 r ⭈ u1 r 2 ⭈ u1 r 3 ⭈ u1 . . . , then r ⭈ S r ⭈ u1 r ⭈ u1 r 2 ⭈ u 1 r 3 ⭈ u 1 . . ., or r ⭈ u 1 r 2 ⭈ u 1 r 3 ⭈ u 1 . . . . Subtract
and solve for S.
Step 5
r
⭈
S u1 r ⭈ u1 + r 2 ⭈ u1 r 3 ⭈ u1 . . .
S r ⭈ u1 r 2 ⭈ u1 r 3 ⭈ u1 . . .
S rS u 1
S
⭈ (1 r) u 1
Subtract.
Factor.
u1
S _____
Divide both sides by (1 r).
1r
Step 6 The partial sums of a geometric sequence will converge to a unique
number S when r is between 1 and 1, or when u1 0.
Read Example B in your book, in which a graph of partial sums is used to
find the sum of a series. Read the example carefully and make sure you
understand the method. Then, read the box after that example, which summarizes
the formula for finding the sum of a convergent infinite geometric series. Note
that a geometric series converges only if ⏐r⏐ 1 or u1 0. Then work through
Example C. Here is another example.
EXAMPLE
Find the sum of the infinite series
∞
130(0.84)n1
n1
䊳
Solution
u1
In this case, r 0.84 and u1 130. Using the formula S ____
1r ,
130 812.5
S ________
1 0.84
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CONDENSED
LE SSO N
9.3
Partial Sums of Geometric Series
In this lesson you will
●
discover a formula for partial sums of geometric series
In Lesson 9.2, you found sums of convergent geometric series. In this lesson, you
will find partial sums of geometric series. Example A in your book shows you
how to use a calculator table or graph to find partial sums of a geometric series.
Read the example carefully.
In Lesson 9.1, you discovered a formula for partial sums of arithmetic series. In
this investigation, you’ll find a formula for partial sums of geometric series.
Investigation: Geometric Series Formula
Work through the investigation in your book. Then check your work against the
results below.
The sequence is defined by u1 180 and un 0.65
heights and partial sum are given in the tables below.
Step 1
Step 2
⭈ un1. The first ten
The scatterplot of the data is shown below.
u
180
180
1
______
___
The long-run value L is given by ____
1 r 1 0.65 0.35 . To find the values
of a and b, substitute the coordinates of the points (1, 180) and (2, 297) into
180
n
Sn ___
0.35 ab to get the system
180 ab
180 ____
0.35
180 ab 2
297 ____
0.35
180
180
___
2
You can rewrite these equations as ab ___
0.35 180 and ab 0.35 297.
___
180
0.35 297
Dividing the second equation by the first gives b _______
___
180 180 0.65.
0.35
180
Substituting 0.65 for b in the first equation gives 0.65a ___
0.35 180.
1
So a 180___
0.35 1
0.65 ___
180
1
1
___
___
⭈ ___
0.65 1800.35 0.65 0.35 . So the equation is
180
180
180
180
___
___
___
n
x
Sn ___
0.35 0.35 (0.65) , or as an exponential function, y 0.35 0.35 (0.65) .
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Lesson 9.3 • Partial Sums of Geometric Series (continued)
u1
u1
____
The equation from Step 2 can be rewritten as Sn ____
1r 1r
u1
u1
Sn _____
(1 r n)
Factor out _____ .
1r
1r
n
u11 r Sn _________
Rewrite the equation.
1r
Step 4 Sn u 1 u 1 ⭈ r + u1 ⭈ r 2 . . . u 1 ⭈ r n1
Step 3
⭈ r n.
r ⭈ Sn u1 ⭈ r u1 ⭈ r 2 . . . u 1 ⭈ r n1 u 1 ⭈ r n
Sn r ⭈ Sn u 1 u 1 ⭈ r n, or u 11 r n
Sn(1 r) u11 r n
u11 r n
Sn _________
1r
Step 5
S10 for the bouncing ball is given by
1801 0.6510
S10 _____________ 507.362
1 0.65
This can be verified on the calculator table.
For the geometric sequence 2, 6, 18, 54, and so on, u1 2 and r 3.
21 310
S10 _________ 59,048
13
Now you have an explicit formula for finding a partial sum of any geometric
series. You need to know only the first term, the common ratio, and the number
of terms. To practice using the formula, work through Examples B and C in your
book. Then read the example below.
EXAMPLE
Find each partial sum.
11
a.
9(2.75)n1
n1
b. 1024 768 576 136.6875
䊳
Solution
a. u1 9 and r 2.75. Use the formula for the partial sum S11.
u11 r 11
91 2.7511
____________ 349,830.5303
S11 __________
1 2.75
(1 r)
b. The first term, u1, is 1024. Each term is three-fourths
the previous term, so r 0.75. Enter u1 1024 and
un 0.75un1 into your calculator and make a table.
The last term given, 136.6875, is u 8. So you need to
find S8. Using the formula,
u11 r 8
10241 0.758
S8 _________
______________ 3685.9375
1 0.75
(1 r)
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