Sequences Quiz

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1. 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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2. Find the limit of $a_n = 2^n \cdot 3^{-n}$.

   (a) $0$
   (b) $1$
   (c) $\infty$
   (d) $\dfrac{2}{\sqrt{3}}$

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3. Find $$\lim_{n \to \infty} \arctan{\sqrt{n+7e^n}}.$$

   (a) $0$
   (b) $\dfrac{\pi}{4}$
   (c) $1$
   (d) $\dfrac{\pi}{2}$

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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. Find the limit, if it exists, of $a_n = \left(\dfrac{n+1}{n}\right)^n$.

   (a) $0$
   (b) $x$
   (c) $\infty$
   (d) $e$

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