Taylor Series Quiz

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1. Find the sum $\sum_{n=2}^\infty \dfrac{\pi^n}{n!}$.

   (a) $e^\pi - 1 - \pi$
   (b) $e^\pi$
   (c) $e^\pi - 1$
   (d) $e^\pi - 2$

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2. Find the Taylor series for $f(x) = 1 + x + x^2$ centered at $a = 2$.

   (a) $1 + x + x^2$
   (b) $1 + 2x$
   (c) $\sum_{n=0}^\infty (x-2)^n$
   (d) $7 + 5(x-2) + (x-2)^2$

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3. Write the Maclaurin series for $f(x) = \frac{1}{x}$ centered at $a = 5$.

   (a) $\sum_{n=0}^\infty \dfrac{(-1)^n(x-5)^n}{5^{n+1}}$
   (b) $\sum_{n=0}^\infty \dfrac{(-1)^n(x-5)^n}{5^n}$
   (c) $\sum_{n=1}^\infty \dfrac{(-1)^n}{x^n}$
   (d) $\sum_{n=0}^\infty \dfrac{(x-5)^n}{5^{n-1}}$

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4. Find a power series (centered at 0) for $f(x) = e^{-3x^3}$.

   (a) $\sum_{n=0}^\infty \dfrac{(-1)^n3^nx^{n^3}}{n!}$
   (b) $\sum_{n=0}^\infty \dfrac{(-1)^n3^nx^{3n}}{n!}$
   (c) $\sum_{n=0}^\infty \dfrac{(-1)^n27^nx^{3n}}{n!}$
   (d) $\sum_{n=0}^\infty \dfrac{(-1)^n3^nx^n}{n!}$

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5. Estimate $\cos{(0.1)}$ using the 4th-degree Taylor polynomial $p_4(x)$ for $\cos{x}$.

   (a) $0.9$
   (b) $0.099$
   (c) $0.995$
   (d) $1.005$

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