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Practice Problems for Module 11

Section 11.9

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos. 
  1. Find the power series representation of \(f(x)\) centered around \(x = 0\) and its radius of convergence. ​
    1. \(\displaystyle f(x)=-\frac{7x^2}{(1+2x)^3}\)​

      \(f(x)=\displaystyle \sum_{n=2}^\infty \frac{-7(-2)^nn(n-1)x^n}{8}\), \(\qquad R=\dfrac{1}{2}\)

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V76


    2. \(\displaystyle f(x)=\ln\left(13-3x^2\right)\)​

      \(f(x)=\displaystyle \ln(13) - \sum_{n=0}^\infty\left(\frac{3}{13}\right)^{n+1}\frac{x^{4n+4}}{n+1}\), \(\qquad R=\displaystyle\sqrt[4]\frac{13}{3}\)

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V77


  2. Evaluate the indefinite integral using a power series: \(\displaystyle \int \! \frac{\arctan\left(5x^4\right)}{x^3} \, dx\)​

    \(\displaystyle \int \! \frac{\arctan\left(5x^4\right)}{x^3} \, dx = \left( \sum_{n=0}^\infty \frac{(-1)^n5^{2n+1}}{(2n+1)(8n+2)}x^{8n+2} \right) + C\)
    Video Errata: The speaker forgot the \(+C\) in the video, but it is necessary because the integral is indefinite.
     

    To see the full video page and find related videos, click the following link.
    WIR 20B M152 V78