Solved Examples: Infinite Geometric Series Convergence & Sum

📚 Quick Study Guide

• 🔢 An infinite geometric series is a series of the form $a + ar + ar^2 + ar^3 + ...$, where $a$ is the first term and $r$ is the common ratio.
• ✅ The series converges (has a finite sum) if the absolute value of the common ratio is less than 1: $|r| < 1$.
• ♾️ If $|r| \geq 1$, the series diverges (does not have a finite sum).
• 🧮 The sum of a convergent infinite geometric series is given by the formula: $S = \frac{a}{1 - r}$.
• 📝 Remember to check the condition for convergence ($|r| < 1$) before applying the sum formula!

Practice Quiz

1. Question 1: For what values of $r$ does the infinite geometric series $\sum_{n=0}^{\infty} r^n$ converge?
   1. A) $r > 0$
   2. B) $r < 1$
   3. C) $|r| < 1$
   4. D) $|r| > 1$
2. Question 2: Find the sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$
   1. A) 1
   2. B) 2
   3. C) $\frac{3}{2}$
   4. D) $\frac{5}{2}$
3. Question 3: Does the infinite geometric series $\sum_{n=0}^{\infty} 2^n$ converge or diverge?
   1. A) Converges
   2. B) Diverges
   3. C) Cannot be determined
   4. D) Converges to 0
4. Question 4: If an infinite geometric series has a first term of 3 and a common ratio of $\frac{1}{3}$, what is its sum?
   1. A) $\frac{9}{2}$
   2. B) $\frac{2}{9}$
   3. C) 9
   4. D) 2
5. Question 5: For what value of $x$ does the series $\sum_{n=0}^{\infty} (x-1)^n$ converge?
   1. A) $0 < x < 2$
   2. B) $x < 2$
   3. C) $x > 0$
   4. D) All real numbers
6. Question 6: The sum of an infinite geometric series is 10, and its first term is 2. What is the common ratio?
   1. A) $\frac{4}{5}$
   2. B) $\frac{5}{4}$
   3. C) $\frac{1}{5}$
   4. D) $\frac{3}{5}$
7. Question 7: Find the sum of the series $\sum_{n=0}^{\infty} \frac{3}{4^n}$.
   1. A) 3
   2. B) 4
   3. C) $\frac{12}{5}$
   4. D) $\frac{5}{12}$