Chapter 5.2

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Write the first four terms of \((a_n)\), where for each natural number \(n\), \(a_n\) is given by \(a_n=5^n.\)

  2. Show that the sequence \((a_n)\) is increasing and the sequence \((b_n)\) is decreasing, where

    1. \(a_n = \tfrac{n+3}{n+5}\)
    2. \(b_n = \tfrac{n}{n^2+1}\)

  3. Identify an example of a sequence \((a_n)\) with the following properties:

    1. \((a_n)\) is strictly increasing and converges to \(2\)
    2. \((a_n)\) is strictly decreasing and converges to \(\frac{1}{4}\)

  4. Use only the Archimedean property of \(\mathbb{R}\) to show that for any positive real number \(\varepsilon\), there is a natural number \(N\) so that if \(n\) is greater than \(N\), then \[|a_n-L|<\varepsilon,\] for the following choices of \((a_n)\) and \(L\):

    1. \(a_n=\frac{9n+2}{3n+4}\) and \(L=3\)
    2. \(a_n=\sqrt{16+\frac{1}{n}}\) and \(L=4\).

  5. Calculate \(\displaystyle\lim_{n\to \infty} \sqrt{100+\tfrac{1}{n}}\). Carefully justify your reasoning.

  6. Take \((a_n)\), \((b_n)\), and \((c_n)\) to be sequences with \[ \lim_{n\to\infty}a_n = 5, \quad \lim_{n\to\infty}b_n = 1, \quad \text{and} \quad \lim_{n\to\infty}c_n = -2.\] Use the limit laws to compute the following: \[\displaystyle \lim_{n\to\infty}\tfrac{(a_n)^3+2b_n}{c_n+\frac{1}{n^5}}.\]

  7. Use the limit laws to determine the following limits:

    1. \(\displaystyle \lim_{n\to\infty}\tfrac{5n}{3n-1}\)
    2. \(\displaystyle \lim_{n\to\infty}\tfrac{n^2+4n}{n^5+n^2+2}\)

  8. Take \((a_n)\) and \((b_n)\) to be sequences with \[\lim_{n\to\infty}a_n=2\quad \text{and} \quad \lim_{n\to\infty}\frac{b_n}{a_n}=18.\] Carefully justify that \((b_n)\) is convergent and calculate its limit.

  9. Calculate \(\displaystyle\lim_{n\to \infty} n\left(\sqrt{100+\tfrac{1}{n}} - 10 \right)\). Carefully justify your reasoning.

  10. Take \(a_n\) to be the sequence that is given by \[a_n = \tfrac{20n^2 + 5n\cos(n)}{4n^2+4}.\] Calculate \(\displaystyle\lim_{n\to \infty} a_n\). Carefully justify your reasoning.

  11. Determine whether the sequence diverges to either infinity or negative infinity:

    1. \(a_n = \left(\frac{1}{8}\right)^{-n}\)
    2. \(a_n = \ln(n+1)\)
    3. \(a_n = -n^5\)

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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