11.4 Infinite Geometric Series
Example 1:
Write out the terms of the infinite geometric series with terms defined by the rule:
S3 = a1 + a2 + a3
USE YOUR CALCULATOR TO HELP COMPLETE THE TABLE.
Y=
⎛ 1⎞
3⎜ ⎟
⎝ 2⎠
x−1
2nd
GRAPH
(Record the values in the table below.)
n
1
2
3
4
an
What happens when you get to n = 7 ?
5
6
7
8
9
Now, estimate the value of the sum of the infinite geometric series by examining the partial sums. Use
your calculator to find the partial sums. A partial sum is the sum of n terms. For example, the partial sum,
S3 = a1 + a2 + a3 would give the sum of the first 3 terms in the series.
The calculator keystrokes below will give you the nth partial sum, or the sum of n terms in the series.
Change the last “n” in the keystrokes to the desired number of terms you wish to add. To pull up the entire
set of keystrokes again, type 2nd ENTER until you get the keystrokes that you need, then cursor left and
change the n to a new value.
2nd
ENTER
2nd
STAT MATH 5:sum(
STAT OPS
⎛ 1⎞
3⎜ ⎟
⎝ 2⎠
5:seq(
ENTER
S1 =
S2 =
S3 =
S4 =
S5 =
S6 =
S7 =
S8 =
S9 =
∞
⎛ 1⎞
∑ 3⎜⎝ 2 ⎟⎠
n=1
n−1
≈
x−1
, X, 1, n, 1
Example 2:
Write out the terms of the infinite geometric series with terms defined by the rule:
an =
n
an
1 n−1
(2)
2
1
2
3
4
5
6
7
8
9
Now, estimate the value of the sum of the infinite geometric series by examining the partial sums.
S1 =
S2 =
S3 =
S4 =
S5 =
S6 =
S7 =
S8 =
S9 =
∞
1
∑ 2 (2)
n−1
≈
n=1
Now for the easy shortcut….
We can determine the exact sum of an infinite geometric series using the formula:
Sum of an Infinite Geometric Series: S =
a1
for r < 1 where r ≠ 0
1− r
Use the formula to find the sum of each infinite geometric series.
Example 3:
100 + 50 + 25 + 12.5 + …
S=
Example 4:
∞
⎛ 1⎞
∑ 5 ⎜⎝ 4 ⎟⎠
n=1
n−1
∞
⎛ 1⎞
∑ 27 ⎜⎝ 3 ⎟⎠
n=0
S=
S=
NOTE: In sigma notation, you need to identify the
the summation is written as above. i.e.
n
∞
∑ a (r )
1
n=0
n
a1 term. This will always be the coefficient in the expression for an when
or
∞
∑ a (r )
n−1
1
n=1
Example 5: Consider the repeating decimal 0.181818……
The infinite repeating decimal is equivalent to 18(.01) + 18(.0001) + 18 (.000001) +…
= .18 + .0018 + .000018 + …
a1 =
S=
Example 6: Write .777… as a fraction.
r=
More practice:
1. Find the sum, if it exists, of each infinite geometric series.
∞
∑ 4(0.3)
i −1
i =1
5 1 1
1
+ + +
+ ...
2 2 10 50
-3 + 6 + (-12) + 24 + …
2. Find the common ratio of an infinite geometric series that has an initial term of 5 and a sum of 12.
3. Write the repeating decimal .121212… as a fraction.