Sequences Quiz

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1. 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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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. 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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4. Find a formula for the $n$th term of the sequence $\{25, -1/5, 1, -5, \ldots\}_{n=1}^\infty$.

   (a) $\dfrac{5}{(-5)^{n-1}}$
   (b) $\dfrac{25}{(-5)^{n-1}}$
   (c) $\dfrac{-5}{5^{n-1}}$
   (d) $\dfrac{-25}{5^{n-1}}$

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5. Let $a_n = \dfrac{3 + \sin{\left(n\right)}}{\sqrt{n}}$. Find $\lim_{n \to \infty} a_n$.

   (a) $0$
   (b) $1$
   (c) $3e$
   (d) $\infty$

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