Algebra 2 Test Questions: Infinite Geometric Series Convergence

📚 Quick Study Guide

• 🔢 An infinite geometric series has 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 the series converges, the sum $S$ is given by the formula: $S = \frac{a}{1 - r}$.
• ⛔ If $|r| \ge 1$, the series diverges (does not have a finite sum).
• 💡 To determine convergence, focus on finding and analyzing the common ratio $r$.

Practice Quiz

1. What condition must the common ratio $r$ satisfy for an infinite geometric series to converge?
   1. $r > 1$
   2. $r < 1$
   3. $|r| < 1$
   4. $|r| > 1$
2. For the series $2 + 1 + \frac{1}{2} + \frac{1}{4} + ...$, does it converge or diverge?
   1. Converges
   2. Diverges
   3. Cannot be determined
   4. None of the above
3. If an infinite geometric series has a first term $a = 5$ and a common ratio $r = \frac{1}{3}$, what is its sum?
   1. $\frac{15}{2}$
   2. $\frac{5}{3}$
   3. $\frac{2}{15}$
   4. $\frac{3}{5}$
4. Determine whether the series $1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + ...$ converges or diverges.
   1. Converges
   2. Diverges
   3. Cannot be determined
   4. None of the above
5. If an infinite geometric series converges to a sum of 10 and has a first term of 2, what is the common ratio?
   1. $\frac{4}{5}$
   2. $\frac{1}{5}$
   3. $\frac{5}{4}$
   4. $\frac{5}{6}$
6. For what values of $x$ does the series $1 + x + x^2 + x^3 + ...$ converge?
   1. $x > 1$
   2. $x < 1$
   3. $|x| < 1$
   4. $|x| > 1$
7. The series $\sum_{n=0}^{\infty} (\frac{2}{3})^n$ converges. What is its sum?
   1. 2
   2. 3
   3. $\frac{2}{3}$
   4. $\frac{3}{2}$