10.2 Working with Geometric Series
Keywords
Geometric Series 等比级数 • Common Ratio 公比
Convergence Condition 收敛条件 • Infinite Sum Formula 无穷和公式
What is a Geometric Series?
A geometric series is a series where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (\(r\)).
Core Concept: Geometric Series (LIM-7.A.3)
A geometric series with first term \(a\) and common ratio \(r\) takes the form:
To find the common ratio \(r\), you can divide any term by its preceding term: \(r = \frac{a_{n+1}}{a_n}\).
Convergence and the Sum Formula
Unlike the Harmonic series, a geometric series only converges if the "steps" get small enough, quickly enough.
Theorem: Convergence of Geometric Series (LIM-7.A.4)
The geometric series \(\sum_{n=0}^{\infty} ar^n\) converges if and only if:
If it converges, the sum \(S\) of the infinite series is:
If \(|r| \geq 1\), the series diverges.
Example: \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\)
Here, the first term \(a = 1\) and the common ratio \(r = 1/2\). Since \(|1/2| < 1\), the series converges.
Extension: Zeno's Paradox
The Greek philosopher Zeno once argued that motion is impossible. To walk across a room, you must first walk half the distance, then half of the remaining distance, and so on. He claimed you would have to complete an infinite number of tasks, which should take an infinite amount of time.
Geometric series solve this paradox! They prove that the sum of an infinite number of distances (or time intervals) can equal a finite value. In our example above, \(1 + 1/2 + 1/4 \dots\) perfectly equals \(2\).