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

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

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3. Find the interval of convergence for $f(x) = \sum_{n=1}^\infty \dfrac{n!(2x-7)^n}{7^n}$.

   (a) $\left[\dfrac{7}{2}, \dfrac{7}{2}\right]$
   (b) $(-\infty, \infty)$
   (c) $(-1, 1)$
   (d) $\left(\dfrac{7}{2}, \dfrac{7}{2}\right)$

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4. 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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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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