Sequences Quiz

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1. Does $a_n = \sqrt{\dfrac{n+1}{9n+1}}$ converge or diverge? If it converges, find the limit.

   (a) Converges to $\dfrac{1}{9}$
   (b) Converges to $\dfrac{1}{3}$
   (c) Converges to $\dfrac{1}{\sqrt{3}}$
   (d) Diverges

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2. Find the $5$th term of $\{3 \cdot 2^n - 4\}_{n=0}^\infty$.

   (a) $92$
   (b) $44$
   (c) $\infty$
   (d) $12$

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3. Find $$\lim_{n \to \infty} \ln{(2n + 1)} - \ln{(n-1)}.$$

   (a) $0$
   (b) $\ln{2}$
   (c) $\infty$
   (d) $-\infty$

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4. Does $a_n = (-1)^n$ converge or diverge? Why?

   (a) Converges by the Monotone Convergence Theorem
   (b) Converges for another reason
   (c) Diverges because it cycles back and forth between $-1$ and $1$
   (d) Diverges to $\pm \infty$

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5. Is $a_n = n + \sin{n}$ bounded? What is $\lim_{n \to \infty} a_n$?

   (a) Bounded between $-1$ and $\dfrac{\pi}{4}$
   (b) Bounded between $0$ and $1$
   (c) Unbounded, and $\lim_{n \to \infty} a_n = \infty$
   (d) Unbounded, and $\lim_{n \to \infty} a_n = -\infty$

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