Taylor Series Quiz

Find the 3rd-order Taylor polynomial $p_3(x)$ for $f(x) = \sqrt{x}$ centered at $a = 1$.

(a) $1 + \frac{1}{2}(x-1) + \frac{1}{8}(x-1)^2 + \frac{1}{16}(x - 1)^3$

(b) $1 + \frac{1}{2}(x-1) + \frac{1}{8}(x-1)^2 + \frac{3}{16}(x - 1)^3$

(c) $\frac{1}{2\sqrt{x}} + \frac{1}{2}$

(d) $1 + \frac{1}{2}(x-1) + \frac{3}{8}(x-1)^2 + \frac{5}{16}(x - 1)^3$

Find the Taylor series for $f(x) = x^3$ centered at $a = 1$.

(a) $(x-1)^3 + 3(x-1)^2 + 3(x-1) + 1$

(b) $\sum_{n=0}^\infty \dbinom{\infty}{n} (x-1)^n$

(c) $x^3$

(d) $3x^2$

Find the Maclaurin series for $\ln{\sqrt{4 - x^2}}$.

(a) $\sum_{n=1}^\infty (-1)^n4^nx^{2n}$

(b) $\frac{\ln{4}}{2} - \frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

(c) $-\frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

(d) $1 - \frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

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) $[-3, 3], p_4(x) = x^2 + \frac{4}{9}x^3 + \frac{5}{27}x^4$

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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