Power Series Quiz

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1. Compute the interval of convergence for $f(x) = \sum_{n=1}^\infty \dfrac{(x-2)^n}{n^3 4^n}$.

   (a) $(-2, 6)$
   (b) $[-2, 6)$
   (c) $(-2, 6]$
   (d) $[-2, 6]$

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2. What is the radius of convergence for the power series $\sum_{n=0}^\infty x^n$?

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

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3. Find a power series representation of $\ln{(1-x)}$ and its radius of convergence.

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

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4. Find the interval of convergence for $f(x) = \sum_{n=1}^\infty \dfrac{(-1)^nx^n(n+4)}{11^n}$.

   (a) $(-11, 11)$
   (b) $[-11, 11)$
   (c) $(-11, 11]$
   (d) $[-11, 11]$

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5. Write $\dfrac{3}{1-x^3}$ as a power series. What is the radius of convergence?

   (a) $\sum_{n=0}^\infty 3x^n, R = 1$
   (b) $\sum_{n=0}^\infty 3x^{3n}, R = 1$
   (c) $\sum_{n=1}^\infty 3x^{3n}, R = 1$
   (d) $\sum_{n=0}^\infty \dfrac{x^{3n}}{3^{3n-1}}, R = 3$

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