Identifying Convergence and Divergence of Geometric Series

Introduction

Key Concepts

Definition of Geometric Series

• A geometric series is an infinite series of the form: $$S = \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \dots$$ where:
  • a is the first term.
  • r is the common ratio between successive terms.

Convergence of Geometric Series

A geometric series converges when the sum approaches a finite limit as the number of terms increases indefinitely. The condition for convergence is: $$|r| < 1$$ Under this condition, the sum of an infinite geometric series is given by: $$S = \frac{a}{1 - r}$$

**Example:** Find the sum of the infinite geometric series where \( a = 5 \) and \( r = \frac{1}{3} \).

Using the formula: $$S = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = \frac{15}{2} = 7.5$$

Divergence of Geometric Series

A geometric series diverges when the sum does not approach a finite limit. The condition for divergence occurs when: $$|r| \geq 1$$ In such cases, the series either increases without bound or oscillates without settling to a particular value.

**Example:** Determine if the series \( S = \sum_{n=0}^{\infty} 2 \cdot 3^n \) converges or diverges.

Here, \( r = 3 \), and since \( |3| \geq 1 \), the series diverges.

Partial Sums of Geometric Series

The partial sum \( S_n \) of the first \( n \) terms of a geometric series is given by: $$S_n = a \frac{1 - r^n}{1 - r}$$ As \( n \) approaches infinity: - If \( |r| < 1 \), \( r^n \) approaches 0, and \( S_n \) approaches \( \frac{a}{1 - r} \). - If \( |r| \geq 1 \), \( S_n \) does not approach a finite limit.

**Example:** Calculate the partial sum \( S_4 \) for the series \( S = 2 + 6 + 18 + 54 + \dots \).

Here, \( a = 2 \) and \( r = 3 \). $$S_4 = 2 \frac{1 - 3^4}{1 - 3} = 2 \frac{1 - 81}{-2} = 2 \frac{-80}{-2} = 80$$

Applications of Geometric Series

• Financial Mathematics: Calculating present and future values of annuities.
• Physics: Analyzing phenomena like radioactive decay and population growth.
• Computer Science: Understanding algorithms with geometric time complexities.

Common Mistakes to Avoid

• Assuming all geometric series converge regardless of the common ratio.
• Incorrectly applying the convergence formula when \( |r| \geq 1 \).
• Miscalculating the partial sums by overlooking the correct formula.

Visual Representation

Graphing a geometric series can provide visual insight into its behavior. For \( |r| < 1 \), the series approaches a horizontal asymptote representing the sum. For \( |r| \geq 1 \), the partial sums will either grow without bound or oscillate indefinitely.

Relation to Other Series

Geometric series serve as a foundational concept for understanding other types of infinite series, such as arithmetic series and p-series. Mastery of geometric series aids in the study of power series and Taylor series expansions.

Infinite Geometric Series Formula Derivation

Deriving the sum formula for an infinite geometric series involves multiplying the series by the common ratio and subtracting:

Let \( S = a + ar + ar^2 + ar^3 + \dots \) \( rS = ar + ar^2 + ar^3 + ar^4 + \dots \) Subtracting the two: $$S - rS = a$$ $$S(1 - r) = a$$ Thus: $$S = \frac{a}{1 - r}$$

Convergence Tests for Geometric Series

While geometric series inherently follow a specific pattern, understanding convergence tests is beneficial:

• Ratio Test: Confirms convergence for \( |r| < 1 \).
• Root Test: Also validates convergence based on \( |r| \).

Example Problems

Problem 1: Determine if the series \( \sum_{n=1}^{\infty} \frac{5}{2^n} \) converges or diverges. If it converges, find its sum.

Solution: Here, \( a = \frac{5}{2} \) and \( r = \frac{1}{2} \). Since \( |r| = \frac{1}{2} < 1 \), the series converges. $$S = \frac{\frac{5}{2}}{1 - \frac{1}{2}} = \frac{\frac{5}{2}}{\frac{1}{2}} = 5$$

Problem 2: Evaluate the sum of the infinite geometric series \( 10 - 10 + 10 - 10 + \dots \).

Solution: Here, \( a = 10 \) and \( r = -1 \). Since \( |r| = 1 \geq 1 \), the series diverges.

Comparison Table

Summary and Key Takeaways