brandon.bell
brandon.bell 6h ago โ€ข 0 views

Algebra 2 Test Questions: Infinite Geometric Series Convergence

Hey there! ๐Ÿ‘‹ Infinite geometric series can seem tricky, but don't worry, we'll break it down. This study guide + quiz will help you master convergence in no time! Let's get started! ๐Ÿค“
๐Ÿงฎ Mathematics

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ryan689 Jan 6, 2026

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข 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.
  • ๐ŸŽฏ The series converges (has a finite sum) if the absolute value of the common ratio is less than 1: $|r| < 1$.
  • ๐Ÿงฎ If the series converges, the sum $S$ is given by the formula: $S = \frac{a}{1 - r}$.
  • โ›” If $|r| \ge 1$, the series diverges (does not have a finite sum).
  • ๐Ÿ’ก 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. $\frac{3}{2}$
Click to see Answers
  1. C
  2. A
  3. A
  4. A
  5. A
  6. C
  7. B

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