Power Series Quiz

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1. Compute the interval of convergence for $\sum_{n=1}^\infty \dfrac{(5x)^n}{n^2}$.

   (a) $\left[\dfrac{-1}{5}, \dfrac{1}{5}\right]$
   (b) $(-\infty, \infty)$
   (c) $[-5, 5]$
   (d) $\left(\dfrac{-1}{5}, \dfrac{1}{5}\right)$

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2. Write $\dfrac{1}{1 - 2x}$ 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. 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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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 the radius of convergence of $\sum_{n=2}^\infty \dfrac{n! x^n}{n^n}$.

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

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