Solved Exercises on Infinite Geometric Series in Algebra 2

Master infinite geometric series: convergence, divergence, applications, and problem-solving techniques through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Infinite Series
Exercise 1
Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + 1/2 + ...
Definition:

Infinite Geometric Series: A geometric series with infinitely many terms that may converge to a finite sum

Convergence: The series converges when |r| < 1, where r is the common ratio

\(S_{\infty} = \frac{a_1}{1 - r} \text{ (when } |r| < 1)\)
Sum Formula for Convergent Infinite Geometric Series
Solution Method:
  1. Identify the first term (a₁) and common ratio (r)
  2. Verify the convergence condition |r| < 1
  3. Apply the infinite series formula
  4. Calculate the sum
Given
a₁=8, r=0.5
Condition
|r| < 1 ✓
Result
S∞=16
Step 1: Identify sequence parameters

a₁ = 8 (first term)

r = 4/8 = 0.5 (common ratio)

Step 2: Check convergence condition

|r| = |0.5| = 0.5 < 1 ✓

Since |r| < 1, the series converges and has a finite sum.

Step 3: Apply the formula

S∞ = a₁/(1 - r)

S∞ = 8/(1 - 0.5)

S∞ = 8/0.5

S∞ = 16

The sum of the infinite series is 16
Final answer:

S∞ = 16

Applied rules:

• Convergence condition: |r| < 1 for infinite series to converge

• Infinite sum formula: S∞ = a₁/(1 - r) when |r| < 1

• Ratio calculation: r = a₂/a₁

Tip 1: Always check the convergence condition before applying the infinite series formula.
Tip 2: If |r| ≥ 1, the infinite series diverges and has no finite sum.
2 Divergent Series
Exercise 2
Does the infinite geometric series 5 + 10 + 20 + 40 + ... converge? If so, find its sum.
Definition:

Divergent Series: A series that does not approach a finite limit as the number of terms increases

\(\text{Diverges when } |r| \geq 1\)
Divergence Condition
Given
a₁=5, r=2
Condition
|r| ≥ 1 ⇒ Diverges
Result
No finite sum
Step 1: Identify the sequence parameters

a₁ = 5 (first term)

r = 10/5 = 2 (common ratio)

Step 2: Check convergence condition

|r| = |2| = 2 > 1

Since |r| ≥ 1, the series diverges.

Step 3: Analyze the behavior

As n increases, the terms grow exponentially: 5, 10, 20, 40, 80, 160, ...

The partial sums grow without bound: 5, 15, 35, 75, 155, ...

The series diverges and has no finite sum
Final answer:

The series diverges; no finite sum exists.

Applied rules:

• Divergence condition: |r| ≥ 1 for infinite series to diverge

• Ratio calculation: r = a₂/a₁

• Behavior analysis: Terms grow exponentially when |r| > 1

Tip 1: A series with |r| ≥ 1 will always diverge.
Tip 2: When |r| = 1, the series also diverges (terms don't approach zero).
3 Negative Ratio Series
Exercise 3
Find the sum of the infinite geometric series: 12 - 4 + 4/3 - 4/9 + 4/27 - ...
Definition:

Alternating Series: A geometric series with negative common ratio, causing terms to alternate in sign

\(S_{\infty} = \frac{a_1}{1 - r} \text{ (when } |r| < 1)\)
Still Applies to Alternating Series
Given
a₁=12, r=-1/3
Condition
|r| < 1 ✓
Result
S∞=9
Step 1: Identify the sequence parameters

a₁ = 12 (first term)

r = -4/12 = -1/3 (common ratio)

Step 2: Check convergence condition

|r| = |-1/3| = 1/3 < 1 ✓

Since |r| < 1, the alternating series converges.

Step 3: Apply the formula

S∞ = a₁/(1 - r)

S∞ = 12/(1 - (-1/3))

S∞ = 12/(1 + 1/3)

S∞ = 12/(4/3)

Step 4: Calculate the sum

S∞ = 12 × 3/4 = 36/4 = 9

The sum of the infinite series is 9
Final answer:

S∞ = 9

Applied rules:

• Alternating series: Still converges if |r| < 1 regardless of sign

• Absolute value: Use |r| for convergence test

• Same formula: S∞ = a₁/(1 - r) works for any r with |r| < 1

Tip 1: Alternating series can still converge if |r| < 1.
Tip 2: Always use absolute value for convergence tests.
Key Formulas and Properties
\(S_{\infty} = \frac{a_1}{1 - r} \text{ (when } |r| < 1)\)
Convergent Infinite Series
\(\text{Diverges when } |r| \geq 1\)
Divergent Infinite Series
Property 1
Convergence
Series converges when |r| < 1
Property 2
Divergence
Series diverges when |r| ≥ 1
Property 3
Limit Behavior
Terms approach zero when |r| < 1
Key Definitions:

Infinite Geometric Series: A geometric series with infinitely many terms

Convergence: When the sum of an infinite series approaches a finite value

Divergence: When the sum of an infinite series does not approach a finite value

Common Ratio: The constant value multiplied to get from one term to the next

Problem-Solving Strategy:
  1. Identify the type: Confirm it's a geometric series
  2. Find parameters: Determine a₁ and r
  3. Check convergence: Verify |r| < 1 for convergence
  4. Apply formula: Use S∞ = a₁/(1 - r) if convergent
  5. State result: Clearly indicate convergence or divergence
Common Errors: Forgetting convergence condition, applying formula to divergent series, sign errors with negative ratios.
Exam Tips: Always check convergence first, memorize the convergence condition, practice with alternating series.
Solution: Exercises 4 to 5
4 Real-world Application
Exercise 4
A ball is dropped from a height of 1 meter. Each time it bounces, it reaches 80% of its previous height. What is the total vertical distance traveled by the ball?
Definition:

Real-world Application: Infinite geometric series model exponential decay processes

Setup
a₁=1, r=0.8
Formula
S∞=a₁/(1-r)
Result
Total distance=9m
Step 1: Analyze the bouncing pattern

Initial drop: 1 meter

First bounce up: 1 × 0.8 = 0.8 meters

First bounce down: 0.8 meters

Second bounce up: 0.8 × 0.8 = 0.64 meters

Second bounce down: 0.64 meters

And so on...

Step 2: Separate the series

Downward distances: 1 + 0.8 + 0.64 + 0.512 + ... (infinite geometric series)

Upward distances: 0.8 + 0.64 + 0.512 + ... (infinite geometric series starting at 0.8)

Step 3: Calculate each series

Downward series: S₁ = 1/(1-0.8) = 1/0.2 = 5 meters

Upward series: S₂ = 0.8/(1-0.8) = 0.8/0.2 = 4 meters

Step 4: Calculate total distance

Total distance = Downward + Upward = 5 + 4 = 9 meters

The ball travels a total distance of 9 meters
Final answer:

Total distance = 9 meters

Applied rules:

• Modeling: Translate physical phenomena into geometric series

• Convergence: Use infinite series when process continues indefinitely

• Separation: Break complex problems into simpler series

Tip 1: Draw a diagram to visualize the bouncing pattern.
Tip 2: Separate upward and downward movements to form distinct series.
5 Finding Unknown Parameters
Exercise 5
An infinite geometric series has a first term of 10 and a sum of 25. Find the common ratio.
Definition:

Inverse Problem: Given the sum and first term, find the common ratio

Given
a₁=10, S∞=25
Formula
r=(S∞-a₁)/S∞
Result
r=0.6
Step 1: Set up the equation

Using the formula: S∞ = a₁/(1 - r)

25 = 10/(1 - r)

Step 2: Solve for r

25(1 - r) = 10

25 - 25r = 10

-25r = 10 - 25

-25r = -15

r = 15/25 = 3/5 = 0.6

Step 3: Verify the solution

Check: S∞ = 10/(1 - 0.6) = 10/0.4 = 25 ✓

Also check convergence: |r| = |0.6| = 0.6 < 1 ✓

Step 4: General formula derivation

Starting with S∞ = a₁/(1 - r)

S∞(1 - r) = a₁

S∞ - S∞r = a₁

S∞r = S∞ - a₁

r = (S∞ - a₁)/S∞

The common ratio is 0.6
Final answer:

Common ratio: r = 0.6

Applied rules:

• Algebraic manipulation: Rearrange formulas to solve for unknown variables

• Verification: Always check that the solution satisfies original conditions

• Convergence check: Ensure |r| < 1 for validity of the formula

Tip 1: When solving for r, isolate it by cross-multiplying.
Tip 2: Derive the general formula r = (S∞ - a₁)/S∞ for future reference.
Comprehensive Guide: Infinite Geometric Series
\(S_{\infty} = \frac{a_1}{1 - r} \text{ (when } |r| < 1)\)
Convergence Formula
\(\text{Diverges when } |r| \geq 1\)
Divergence Condition
Key definitions:

Infinite Geometric Series: A geometric series with infinitely many terms: a₁ + a₁r + a₁r² + a₁r³ + ...

Convergence: When the sum of an infinite series approaches a finite limit

Divergence: When the sum of an infinite series does not approach a finite limit

Common Ratio: The constant value multiplied to get from one term to the next

Complete methodology:
  1. Identify the series: Confirm it's geometric by checking for constant ratio
  2. Find parameters: Determine a₁ (first term) and r (common ratio)
  3. Check convergence: Verify |r| < 1
  4. Apply appropriate formula: Use S∞ = a₁/(1 - r) if convergent
  5. Solve systematically: Apply algebraic techniques
  6. Verify results: Check convergence condition and arithmetic
Tip 1: Always verify that |r| < 1 before applying the convergence formula.
Tip 2: Remember that convergence only occurs when |r| < 1, regardless of the sign of r.
Tip 3: For alternating series (negative r), still check |r| < 1 for convergence.
Tip 4: Infinite geometric series model exponential decay and growth processes.
Common errors: Forgetting convergence condition, applying formula to divergent series, calculation errors with fractions.
Exam preparation: Memorize convergence condition, practice with various ratios including negative ones, work on real-world applications.
Essential rules to know:

• Convergence condition: |r| < 1

• Convergent sum: S∞ = a₁/(1 - r)

• Divergence: |r| ≥ 1

• Ratio: r = aₙ/aₙ₋₁

Visual Understanding: Convergence Behavior
Exercise 6: Convergence Analysis
Compare the partial sums of two series:
Convergent: 1 + 1/2 + 1/4 + 1/8 + ... (r = 1/2)
Divergent: 1 + 2 + 4 + 8 + ... (r = 2)

Analysis: The visualization shows how convergent series approach their limit while divergent series grow without bound.

  • Convergent series approaches limit of 2 as terms approach zero
  • Divergent series grows exponentially as terms increase
  • Convergence occurs when |r| < 1, divergence when |r| ≥ 1

Questions & Answers

Question: I don't understand why some infinite series converge to a finite sum while others don't. Can you explain?

Answer: The key is how quickly the terms approach zero:

  • Convergent series (|r| < 1): Each term gets smaller and smaller, approaching zero. The terms decrease fast enough that their sum approaches a finite value.
  • Divergent series (|r| ≥ 1): Terms either stay the same size or grow larger, so the sum keeps increasing without bound.

Think of it like a race to zero: if terms reach zero fast enough (when |r| < 1), the sum is finite. If they don't reach zero quickly enough (when |r| ≥ 1), the sum is infinite.

Example: 1 + 1/2 + 1/4 + 1/8 + ... converges to 2 because terms halve each time, approaching zero rapidly.

Question: What happens when r = 1 or r = -1? Do those series converge?

Answer: Neither converges:

  • When r = 1: Series becomes a₁ + a₁ + a₁ + ... = a₁ + a₁ + a₁ + ... which clearly diverges to infinity.
  • When r = -1: Series becomes a₁ - a₁ + a₁ - a₁ + ... which oscillates between a₁ and 0, never approaching a single limit.

Both cases fail the convergence condition |r| < 1. For convergence, we need |r| to be strictly less than 1, not equal to 1.

This is why the convergence condition is |r| < 1, not |r| ≤ 1.

Question: What are some real-world applications of infinite geometric series? Why is it important to learn this?

Answer: Infinite geometric series have numerous practical applications:

  • Finance: Calculating present value of perpetuities, mortgage payments, compound interest
  • Physics: Modeling radioactive decay, light reflection, heat dissipation
  • Engineering: Signal processing, control systems, feedback analysis
  • Economics: Multiplier effects in economic models
  • Computer Science: Algorithm analysis for recursive processes

Learning infinite geometric series develops understanding of limits, convergence, and mathematical modeling of continuous processes. These concepts are fundamental to calculus and advanced mathematics.

The ability to model processes that continue indefinitely is essential for many STEM fields.

Question: I sometimes confuse infinite geometric series with finite geometric series. How do I keep them straight?

Answer: Here's how to distinguish them:

Feature Finite Geometric Series Infinite Geometric Series
Number of terms Fixed number n Infinitely many
Formula Sₙ = a₁(rⁿ - 1)/(r - 1) S∞ = a₁/(1 - r)
Condition r ≠ 1 |r| < 1
Result Always finite Finite if |r| < 1, infinite otherwise

Remember: finite series always have a sum, infinite series only converge when |r| < 1.

Question: Can infinite geometric series be used to represent repeating decimals? How?

Answer: Yes! Repeating decimals can be expressed as infinite geometric series:

Example: Convert 0.333... to fraction

  • 0.333... = 0.3 + 0.03 + 0.003 + ...
  • = 3/10 + 3/100 + 3/1000 + ...
  • = 3/10 + (3/10)(1/10) + (3/10)(1/10)² + ...
  • This is a geometric series with a₁ = 3/10 and r = 1/10
  • S∞ = (3/10)/(1 - 1/10) = (3/10)/(9/10) = 3/9 = 1/3

This method works for any repeating decimal by expressing it as a geometric series with |r| < 1.

This is a powerful application showing how infinite series connect to basic number representation.