franceswalker1985
franceswalker1985 2d ago โ€ข 0 views

Test Your Knowledge: Convergence and Divergence of Sequences

Hey there! ๐Ÿ‘‹ Ready to test your knowledge on the convergence and divergence of sequences? Let's dive in with a quick study guide, followed by a practice quiz to solidify your understanding. Good luck! ๐Ÿ€
๐Ÿงฎ Mathematics

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heather_wilson Dec 27, 2025

๐Ÿ“š Quick Study Guide

    ๐Ÿ” A sequence is a list of numbers in a specific order, often defined by a formula. ๐Ÿ“ˆ Convergence means the sequence approaches a finite limit as the index approaches infinity. ๐Ÿ“‰ Divergence means the sequence does not approach a finite limit; it may oscillate or increase/decrease without bound. ๐Ÿ”ข Key limits to remember:
  • $lim_{n \to \infty} \frac{1}{n^p} = 0$ for $p > 0$
  • $lim_{n \to \infty} r^n = 0$ for $|r| < 1$
  • ๐Ÿงฎ L'Hรดpital's Rule can be used to evaluate limits of sequences when they take indeterminate forms. ๐Ÿงช Common convergence tests include the ratio test, root test, and comparison test. ๐Ÿ’ก Monotonic sequences (either increasing or decreasing) that are bounded are guaranteed to converge.

Practice Quiz

  1. What does it mean for a sequence to converge?
    1. It oscillates between two values.
    2. It approaches a finite limit.
    3. It increases without bound.
    4. It decreases without bound.
  2. Which of the following sequences converges?
    1. $a_n = (-1)^n$
    2. $a_n = n^2$
    3. $a_n = \frac{1}{n}$
    4. $a_n = sin(n)$
  3. The sequence $a_n = \frac{n}{n+1}$ converges to which value?
    1. 0
    2. 1
    3. -1
    4. $\infty$
  4. Which test is most suitable for determining the convergence of the sequence $a_n = \frac{n!}{n^n}$?
    1. Comparison Test
    2. Ratio Test
    3. Root Test
    4. Integral Test
  5. If a sequence is monotonically increasing and bounded above, it must:
    1. Diverge to infinity.
    2. Diverge to negative infinity.
    3. Converge.
    4. Oscillate.
  6. What is the limit of the sequence $a_n = (1 + \frac{1}{n})^n$ as $n$ approaches infinity?
    1. 0
    2. 1
    3. $e$
    4. $\infty$
  7. Which of the following statements is true about a divergent sequence?
    1. It always approaches a finite limit.
    2. It never oscillates.
    3. It does not approach a finite limit.
    4. It is always bounded.
Click to see Answers
  1. B
  2. C
  3. B
  4. B
  5. C
  6. C
  7. C

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