12-5
Mathematical Induction and
Infinite Geometric Series
LEARNING GOALS FOR LESSON 12.5
12.5.1 Determine if a geometric series diverges or converges
12.5.2 Find the sum of a geometric series
12.5.3 Write repeating decimals as fractions
.
You can find the sums of some infinite geometric series.
Consider the two infinite geometric series below:
Notice that the series Sn has a common ratio of ____ and the
partial sums get closer and closer to ____ as n increases.
When |r|____1 and the partial sum approaches a fixed number, the
series is said to ________________.
The number that the partial sums approaches is called a _______.
For the series Rn, the opposite applies. Its common ratio is ___,
and its partial sums increase toward _____.
When |r| ____ 1 and the partial sum does not approach a fixed
number, the series is said to ________________.
12-5
Mathematical Induction and
Infinite Geometric Series
Ex. 1: Finding Convergent or Divergent Series
LG 12.5.1
Determine whether each geometric series converges or
diverges.
A. 10 + 1 + 0.1 + 0.01 + ...
B. 4 + 12 + 36 + 108 + ...
C.
D. 32 + 16 + 8 + 4 + 2 + …
Ex. 2: Find the Sums of Infinite Geometric Series
LG 12.5.2
Find the sum of the infinite geometric series, if it exists.
A. 1 – 0.2 + 0.04 – 0.008 + ...
B.
12-5
Mathematical Induction and
Infinite Geometric Series
✔ It Out! Ex. 2: Find the Sums of Infinite Geometric Series
LG 12.5.2
Find the sum of the infinite geometric series, if it exists.
A.
B.
You can use infinite series to write a repeating decimal as a fraction.
Ex. 3: Writing Repeating Decimals as Fractions
A. Write 0.63 as a fraction in simplest form.
Step 1 Write the repeating
decimal as an infinite geometric
series.
Step 2 Find the common ratio.
Step 3 Find the sum.
B. Write 0.111… as a fraction in simplest form.
LG 12.5.3