Defining Geometric Series

Introduction

Key Concepts

Definition of a Geometric Series

A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant ratio, denoted as \( r \). Mathematically, a geometric series can be expressed as: $$ S_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1} $$ where:

• Sₙ is the sum of the first n terms.
• a is the first term.
• r is the common ratio.

If the series continues indefinitely, it is referred to as an infinite geometric series: $$ S = a + ar + ar^2 + ar^3 + \dots $$

Common Ratio (\( r \))

The common ratio is the factor by which consecutive terms of the sequence multiply. It determines the behavior of the series:

• If \( |r| < 1 \), the infinite series converges.
• If \( |r| \geq 1 \), the infinite series diverges.

For example, in the sequence 3, 6, 12, 24, ..., the common ratio \( r = 2 \).

Sum of Finite Geometric Series

The sum of the first n terms of a geometric series can be calculated using the formula: $$ S_n = a \cdot \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 $$ This formula provides a quick way to sum a finite number of terms without performing term-by-term addition.

Sum of Infinite Geometric Series

An infinite geometric series converges to a sum if the absolute value of the common ratio is less than one. The sum of an infinite geometric series is given by: $$ S = \frac{a}{1 - r}, \quad \text{for } |r| < 1 $$ This convergence occurs because the terms become progressively smaller, approaching zero.

Convergence and Divergence

Whether a geometric series converges or diverges depends on the common ratio:

• Convergent Series: If \( |r| < 1 \), the series converges to \( \frac{a}{1 - r} \).
• Divergent Series: If \( |r| \geq 1 \), the series does not approach a finite limit.

For instance, the series \( 2 + 1 + 0.5 + 0.25 + \dots \) with \( r = 0.5 \) converges to 4.

Applications of Geometric Series

Geometric series have numerous applications in various fields:

• Finance: Modeling compound interest and calculating present and future values.
• Physics: Analyzing phenomena like radioactive decay and alternating current circuits.
• Computer Science: Designing algorithms and understanding algorithmic complexity.
• Economics: Studying growth models and depreciation of assets.

Example Problems

• Example 1: Find the sum of the first 5 terms of the geometric series where \( a = 3 \) and \( r = 2 \).
  Solution: $$ S_5 = 3 \cdot \frac{1 - 2^5}{1 - 2} = 3 \cdot \frac{1 - 32}{-1} = 3 \cdot 31 = 93 $$
• Example 2: Determine the sum of the infinite geometric series \( 5 + 2.5 + 1.25 + \dots \)
  Solution: Here, \( a = 5 \) and \( r = 0.5 \). $$ S = \frac{5}{1 - 0.5} = \frac{5}{0.5} = 10 $$
• Example 3: Does the series \( 1 - 3 + 9 - 27 + \dots \) converge or diverge?
  Solution: The common ratio \( r = -3 \), and \( |r| = 3 \geq 1 \). Therefore, the series diverges.

Derivation of the Sum Formulas

• Sum of Finite Series: To derive the sum of a finite geometric series, consider: $$ S_n = a + ar + ar^2 + \dots + ar^{n-1} $$ Multiply both sides by \( r \): $$ rS_n = ar + ar^2 + \dots + ar^{n} $$ Subtract the second equation from the first: $$ S_n - rS_n = a - ar^{n} $$ Factor out \( S_n \) and solve for it: $$ S_n(1 - r) = a(1 - r^{n}) $$ $$ S_n = a \cdot \frac{1 - r^{n}}{1 - r} $$
• Sum of Infinite Series: Taking the limit as \( n \) approaches infinity for \( |r| < 1 \): $$ S = \lim_{n \to \infty} S_n = \lim_{n \to \infty} a \cdot \frac{1 - r^{n}}{1 - r} $$ Since \( |r| < 1 \), \( r^{n} \to 0 \): $$ S = \frac{a}{1 - r} $$

Properties of Geometric Series

• Closed-Form Expression: Geometric series have a simple closed-form expression for their sum, which simplifies calculations.
• Self-Similarity: Removing the first term from a geometric series results in another geometric series with the same ratio.
• Infinite Series Behavior: The sum of an infinite geometric series depends solely on the common ratio, determining convergence or divergence.
• Relationship with Exponents: Geometric series are closely related to exponential functions due to the constant ratio between terms.

Advanced Topics

• Power Series: Geometric series are a special case of power series where each term is a power of a common ratio.
• Radius of Convergence: In the context of power series, the geometric series helps determine the radius within which the series converges.
• Application in Calculus: Geometric series aid in solving differential equations and evaluating integrals involving infinite series.
• Generating Functions: Geometric series are used in generating functions to encode sequences and solve combinatorial problems.

Comparison Table

Aspect	Finite Geometric Series	Infinite Geometric Series
Definition	Sum of a finite number of terms in a geometric sequence.	Sum of infinitely many terms in a geometric sequence.
Sum Formula	$S_n = a \cdot \frac{1 - r^n}{1 - r}$	$S = \frac{a}{1 - r}$, provided $|r| < 1$
Convergence	Always converges as it has a finite number of terms.	Converges only if $|r| < 1$.
Applications	Calculating total payments in loan amortization.	Modeling compound interest over an infinite period.
Example	$3 + 6 + 12 + 24$	$5 + 2.5 + 1.25 + \dots$

Summary and Key Takeaways