Taylor Series Quiz

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1. Let $f(x) = \sum_{n=1}^\infty \dfrac{(n+2)x^{n+1}}{3^n}$. Find the interval of convergence for $f(x)$. Then find the fourth-order Taylor polynomial, $p_4(x)$ (centered at $a = 0$), for $f(x)$.

   (a) $[-3, 3], p_4(x) = \sum_{n=1}^\infty \dfrac{(n+1)(n+2)x^n}{3^n}$
   (b) $(-3, 3), p_4(x) = 0$
   (c) $(-3, 3), p_4(x) = x^2 + \frac{4}{9}x^3 + \frac{5}{27}x^4$
   (d) $[-3, 3], p_4(x) = x^2 + \frac{4}{9}x^3 + \frac{5}{27}x^4$

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

   (a) $(x-1)^2 - 2(x-1) + 1$
   (b) $(x-1)^2$
   (c) $(x-1)^2 - 1$
   (d) $(x-1)^2 - 2(x-1)$

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3. Find the Taylor series centered at $a = 1$ for $f(x) = e^{2x}$. What is the radius of convergence?

   (a) $\sum_{n=0}^\infty \dfrac{2^ne^2(x-1)^n}{n!}, R = \infty$
   (b) $\sum_{n=0}^\infty \dfrac{2^ne^2(x-1)^n}{n!}, R = 1$
   (c) $\sum_{n=0}^\infty \dfrac{(x-1)^n}{n!}, R = \infty$
   (d) $\sum_{n=0}^\infty \dfrac{2^n(x-1)^n}{n!}, R = \infty$

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4. Find the 3rd order Taylor polynomial for $f(x) = \sqrt{x+2}$ centered at $a = 7$.

   (a) $p_3(x) = \sqrt{x-7} + 3$
   (b) $p_3(x) = 3 + \frac{1}{6}(x-7) - \frac{1}{216}(x-7)^2 + \frac{1}{3888}(x-7)^3$
   (c) $p_3(x) = 3 + \frac{1}{6}(x-7) - \frac{1}{108}(x-7)^2 + \frac{1}{648}(x-7)^3$
   (d) $p_3(x) = \frac{1}{2\sqrt{x+2}} + \frac{1}{3}$

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5. Let $\sum_{n=0}^\infty c_nx^n$ be the Maclaurin series for $x\sin{\left(\frac{x}{3}\right)}$. Find $c_4$.

   (a) $\dfrac{-1}{81}$
   (b) $\dfrac{-1}{162}$
   (c) $\dfrac{1}{162}$
   (d) $0$

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