Chapter 5.4

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Evaluate the following finite series: \(\sum\limits_{k=10}^{50}(3k-2).\)
  2. Evaluate the following finite series: \(\sum\limits_{k=1}^{40}\left(\tfrac{1}{k+1}-\tfrac{1}{k+2}\right).\)
  3. Evaluate the following finite series: \(\sum\limits_{k=2}^{40}\tfrac{1}{k^2+4k+3}.\)
  4. Evaluate the following finite series: \(\sum\limits_{k=3}^{50}\left(\tfrac{1}{2}\right)^k.\)
  5. Evaluate the following series: \(\sum\limits_{k=2}^{\infty}\tfrac{1}{k^2+4k+3}.\)
  6. Evaluate the following series: \(\sum\limits_{k=3}^{\infty}\left(\tfrac{1}{2}\right)^k.\)
  7. Show that the following series diverges: \(\sum\limits_{n=1}^{\infty}\tfrac{n^2+3}{4n^2+n+1}.\)
  8. Use the comparison test to determine the convergence or divergence of these series:
    1. \(\sum\limits_{n=1}^{\infty}\tfrac{n+1}{n^3+2n+1}\)
    2. \(\sum\limits_{n=1}^{\infty}\tfrac{n}{\sqrt{4n^2+2}}\)
    3. \(\sum\limits_{n=1}^{\infty}\tfrac{3^n+2n}{10^n+100}\)
  9. Use the limit comparison test to determine the convergence or divergence of these series:
    1. \(\sum\limits_{n=1}^{\infty}\tfrac{n+1}{n^3+2n+1}\)
    2. \(\sum\limits_{n=1}^{\infty}\tfrac{n}{\sqrt{4n^2+2}}\)
    3. \(\sum\limits_{n=1}^{\infty}n^2\sin\left(\frac{1}{n^3}\right)\)
    4. \(\sum\limits_{n=1}^{\infty}\tfrac{3^n+2n}{10^n+100}\)
  10. Use the alternating series test to determine the convergence of the following series and then determine if they are conditionally convergent or absolutely convergent:
    1. \(\sum\limits_{n=1}^{\infty}\tfrac{(-1)^n}{n^2+1}\)
    2. \(\sum\limits_{n=1}^{\infty}\tfrac{(-1)^n}{e^n}\)
    3. \(\sum\limits_{n=1}^{\infty}\tfrac{1}{\sqrt{3n^2+1}}\)
  11. Use the ratio test to determine the convergence of the following series: \(\sum\limits_{n=1}^{\infty}\tfrac{3^n}{n!}.\)
  12. Use the root test to determine the convergence of the following series: \(\sum\limits_{n=1}^{\infty}\tfrac{3^n}{2^n}.\)

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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