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Practice Problems for Module 9

Section 11.4

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. Use the Comparison Test to determine if the series converges or diverges.
    1. \(\displaystyle \sum_{n=5}^\infty \frac{7^n-8}{9+5^n}\)  

      Diverges
      Video Errata: The presenter should have shown that 9 + 5n is less than something (not greater than) since we are dividing by it. Rather than \(9 + 5n ≥ 5n\), he should have shown \(9+5n ≤2·5n\). This follows from \(9<5^2 ≤ 5^n\) for \(n≥2\) and so \(9+5^n <5^n +5^n =2·5^n\). Note the final answer is the same in that the series diverges, but with \(1/4\) in front instead of \(1/2\).
       

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V55


    2. \(\displaystyle \sum_{n=1}^\infty \frac{\ln(n)}{\sqrt{n}}\) ​

      Diverges

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V56


    3. \(\displaystyle \sum_{n=1}^\infty \frac{n\ln(n)}{2n^5+\cos(n)}\)  

      Converges
      Video Errata: The \(\cos(1)\) should be \(\cos(n)\). The argument still works.
       

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V57


    4. \(\displaystyle \sum_{n=1}^\infty \frac{2n^3e^{-5n}+1}{7n^6+3n}\)  

      Converges

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V58


  2. ​Use the Limit Comparison Test to determine if the series converges or diverges.
    1. \(\displaystyle \sum_{n=2}^\infty \frac{9n^3+2}{7n^4-\sin(5n)}\)  

      Diverges

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V59


    2.  \(\displaystyle \sum_{n=3}^\infty \frac{\sqrt{5n-6}}{3n^2+1}\)  

      Converges

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V60


    3. \(\displaystyle \sum_{n=2}^\infty \frac{7n^6+n^4}{\sqrt[3]{5n^2-1+n^{15}}}\)  

      Diverges

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V61