Calculus: Geometric Series
Overview
A geometric series is one of the most important infinite series in calculus. It appears
in finance, physics, computer science, and many real-world growth/decay models. These
notes explain how to recognize a geometric series, how to determine whether it converges or
diverges, and how to compute its sum when it converges.
Recognizing a Geometric Series
A geometric series has the form:
∞
X
arn
n=0
where:
• a is the first term (a constant),
• r is the common ratio (also a constant).
Each term is obtained by multiplying the previous term by r.
Example
∞
X
1 · 2n = 1 + 2 + 4 + 8 + 16 + . . .
n=0
Here:
a = 1, r = 2.
Convergence and Divergence
A geometric series behaves very differently depending on the value of r.
1
Overview
A geometric series is one of the most important infinite series in calculus. It appears
in finance, physics, computer science, and many real-world growth/decay models. These
notes explain how to recognize a geometric series, how to determine whether it converges or
diverges, and how to compute its sum when it converges.
Recognizing a Geometric Series
A geometric series has the form:
∞
X
arn
n=0
where:
• a is the first term (a constant),
• r is the common ratio (also a constant).
Each term is obtained by multiplying the previous term by r.
Example
∞
X
1 · 2n = 1 + 2 + 4 + 8 + 16 + . . .
n=0
Here:
a = 1, r = 2.
Convergence and Divergence
A geometric series behaves very differently depending on the value of r.
1
