Lesson 1.5: Infinite Geometric Series
Specific Outcome: Analyze geometric sequences and series to solve problems. (Relations and Functions ~ 10)

Infinite Geometric series is a geometric series that has an infinite number of terms, and the series has no last term.

Types of Series:
 A Convergent Series is a series with an infinite number of terms, in which the sequence of partial sums
approaches a fixed value and the common ratio is a number between -1 and 1 (−1 < 𝑟 < 1). For
example, 1 + ½ + ¼ + …
 A Divergent Series is a series with infinite number of terms, in which the sequence of partial sums does
not approach a fixed value and the common ratio has to be either
r < -1 or r > 1. For example, 2 + 4 +
8+…
Example 1: State whether each infinite geometric series is convergent or divergent.
1
1
1
2
a) 1 - 3 + 9 - …
b) 2 – 1 + - …
c) 2 – 4 + 8 - …
d) 4 + 8 + 16 + …
2
2 2
2 3
Example 2: What is the sum of the infinite geometric series below? 5 + 5 (3) + 5 (3) + 5 (3) + …
Example 3: The first term of an infinite geometric series is −8, and its sum is −
40
.
3
What is the common
ratio? Write the first four terms of the series.
Example 4: Each side of an equilateral triangle has length of 1 cm. The midpoints of the sides are joined
to form an inscribed equilateral triangle. Then, the midpoints of the sides of that triangle are joined to
form another triangle. If this process continues forever, what is the sum of the perimeters of the
triangles?
Example 5: A hot air balloon rises 25 m in its first minute of flight. Suppose that each succeeding minute
the balloon rises only 80% as high as the previous minute. What would be the balloon’s maximum
altitude?
Practice Questions: Page 63: # 2, 6, 8, 15