Math Lesson 8.4: Finding Sums of Infinite Geometric Series

Primary Learning Objectives for Infinite Geometric Series

• The student will learn to find the partial sums of infinite geometric series.

• The student will learn to determine the final sums of infinite geometric series.

Theoretical Foundation: From Finite to Infinite Series

• The study of infinite geometric series begins with the established formula for the sum of a finite geometric series, defined as:     - $S_n = \frac{a_1(1 - r^n)}{1 - r}$

• To understand infinite series, one must analyze the behavior of the sum ($S$) as common terms ($n$) approach infinity ($n \rightarrow \infty$).

Convergence and Divergence Requirements

• Case 1: Common ratio is equal to one ($r = 1$)     - Example: $2 + 2 + 2 + \dots$     - Result: The sum ($S$) approaches infinity ($S \rightarrow \infty$).     - Conclusion: The sum is undefined.

• Case 2: Common ratio is greater than one (r > 1)     - Example: $2 + 4 + 8 + \dots$     - Result: The sum ($S$) approaches infinity ($S \rightarrow \infty$).     - Conclusion: The sum is undefined.

• Case 3: Absolute value of common ratio is less than one (|r| < 1)     - Example: $2 + 1 + \frac{1}{2} + \dots$     - Calculation of term behavior: If $r = \frac{1}{2}$, as $n \rightarrow \infty$, the expression $(\frac{1}{2})^n = 2^{-n}$ approaches zero ($0$).

Derivation and Formula for the Sum of an Infinite Geometric Series

• When the condition |r| < 1 is met, the finite sum formula is modified by substituting the limit of the term $r^n$ (which is $0$) as $n$ reaches infinity:     - $S = \frac{a_1(1 - 0)}{1 - r}$     - $S = \frac{1 \times a_1}{1 - r}$

• The Infinite Geometric Series Sum Rule:     - $S = \frac{a_1}{1 - r}$

• Requirement for use: The common ratio must satisfy the condition |r| < 1.

Representing Infinite Series in Summation (Sigma) Notation

• An infinite series such as $2 + 1 + \frac{1}{2} + \dots$ can be expressed in summation notation using the following parameters:     - First term ($a_1$) = $2$     - Common ratio ($r$) = $\frac{1}{2}$

• General form: $\sum_{n=1}^{\infty} a_1 r^{n-1}$

• Applied form for this series: $\sum_{n=1}^{\infty} 2(\frac{1}{2})^{n-1}$

• Evaluation with Sum Rule:     - $a_1 = 2$, $r = \frac{1}{2}$     - $S = \frac{2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4$

Example 1: Evaluating a Sum in Sigma Notation

• Given Series: $\sum_{j=1}^{\infty} 5(\frac{4}{5})^{j-1}$

• Identification:     - The series is an infinite geometric series because the upper limit is $\infty$.     - $a_1 = 5$     - $r = \frac{4}{5}$

• Condition Check: Since |\frac{4}{5}| < 1, the sum exists.

• Calculation:     - $S = \frac{a_1}{1 - r}$     - $S = \frac{5}{1 - \frac{4}{5}}$     - $S = \frac{5}{\frac{1}{5}}$     - $S = 25$

Example 2: Converting Repeating Decimals to Fractions

• Goal: Write the repeating decimal $x = 0.\overline{24}$ using summation notation and find its sum.

• Step 1: Expand the repeating decimal as a series.     - $x = 0.242424\dots$     - $x = 0.24 + 0.0024 + 0.000024 + \dots$

• Step 2: Identify series parameters.     - First term ($a_1$) = $0.24 = \frac{24}{100}$     - Common ratio ($r$) = $0.01 = \frac{1}{100}$

• Step 3: Write in Summation Notation.     - $\sum_{n=1}^{\infty} 0.24(\frac{1}{100})^{n-1}$

• Step 4: Find the sum.     - $S = \frac{a_1}{1 - r}$     - $S = \frac{0.24}{1 - 0.01} = \frac{0.24}{0.99}$     - $S = \frac{24}{99}$     - Simplification: Dividing both numerator and denominator by $3$ yields $x = \frac{8}{33}$.

Supplemental Geometric Application

• Task: Find the surface area of the composite figure provided in the diagram.

• Composite Dimensions:     - Upper rectangular block: base length = $4\,cm$, height segment = $3\,cm$.     - Base rectangular block: total base width = $10\,cm$, height height segment = $5\,cm$.