Power Series Quiz

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1. Find the interval of convergence of $\sum_{n=1}^\infty \dfrac{2^n(3x-4)^n}{5^n \cdot 6n}$.

   (a) $(2, 6)$
   (b) $[3, 5)$
   (c) $\left[\dfrac{1}{2}, \dfrac{13}{6}\right)$
   (d) $(-\infty, \infty)$

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2. What is the radius of convergence for the series $\sum_{n=0}^\infty (3x)^n$?

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

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3. 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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4. Compute the interval of convergence for $\sum_{n=1}^\infty \dfrac{(-x)^n(n+5)}{3^n}$.

   (a) $(-3, 3]$
   (b) $[-3, 3)$
   (c) $(-\infty, \infty)$
   (d) $(-3, 3)$

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5. 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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