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. Write $\dfrac{1}{1 - x^2}$ as a power series.

   (a) $\sum_{n=0}^\infty x^{2n}$
   (b) $\sum_{n=0}^\infty 2^{2n}x^{2n}$
   (c) $\sum_{n=0}^\infty 2^nx^n$
   (d) $\sum_{n=0}^\infty x^n$

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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 of $\sum_{n=0}^\infty \dfrac{(x-1)^n}{n+5}$.

   (a) $(0, 2)$
   (b) $[0, 2)$
   (c) $(0, 2]$
   (d) $[0, 2]$

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

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

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