\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 5.7 Calculus 2

Comparison Tests
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5.7 Comparison Tests

5.7.1 Direct Comparison Test

  • Following is a list of sequence formulas ordered from larger to smaller (for sufficiently large \(n\)).
    • \(n^n\)
    • \(n!\)
    • \(b^n\) where \(b>1\) (such as \(2^n,e^n,10^n\)…)
    • \(n^p\) where \(p>0\) (such as \(\sqrt{n},n,n^4\)…)
    • \(\log_b n\) where \(b>1\) (such as \(\log_{10}(n),\ln(n),\log_2(n)\)…)
    • any positive constant
  • Example Show that \(\frac{m^3+7}{4^m+5}\leq 2(\frac{1}{2})^m\) for sufficiently large values of \(m\).
  • Suppose \(\sum_{n=N}^\infty a_n\) is a series with non-negative terms.
    • If there exists a convergent series \(\sum_{n=M}^\infty b_n\) with non-negative terms where \(a_n\leq b_n\) for sufficiently large \(n\), then \(\sum_{n=N}^\infty a_n\) converges as well.
    • If there exists a divergent series \(\sum_{n=M}^\infty b_n\) with non-negative terms where \(a_n\geq b_n\) for sufficiently large \(n\), then \(\sum_{n=N}^\infty a_n\) diverges as well.
  • Example Show that \(\sum_{m=0}^\infty \frac{m^3+7}{4^m+5}\) converges by comparing with the series \(\sum_{m=0}^\infty 2(\frac{1}{2})^m\).
  • Example Does \(\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}\) converge or diverge?
  • Example Does \(\sum_{k=3}^\infty\frac{e^k}{e^{2k}-1}\) converge or diverge?
  • Example Does \(\sum_{m=2}^\infty(m\ln m)^{-1/2}\) converge or diverge?

5.7.2 Limit Comparison Test

  • Suppose \(\sum_{n=N}^\infty a_n\) is a series with non-negative terms. If there exists a series \(\sum_{n=M}^\infty b_n\) with non-negative terms where \(0<\lim_{n\to\infty}\frac{a_n}{b_n}<\infty\), then either both series converge or both series diverge.
  • Example Does \(\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}\) converge or diverge?
  • Example Does \(\sum_{i=0}^\infty\frac{i^2+3i+7}{3i^4+2i^2+5}\) converge or diverge?
  • Example Does \(\sum_{n=42}^\infty\frac{2^n+5^n}{3^n+4^n}\) converge or diverge?

Review Exercises

  1. Does \(\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}\) converge or diverge?
  2. Does \(\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}\) converge or diverge?
  3. Does \(\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}\) converge or diverge?
  4. Does \(\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}\) converge or diverge?
  5. Does \(\sum_{m=4}^\infty\frac{1}{\ln m}\) converge or diverge?
  6. Does \(\sum_{n=4}^\infty\frac{5}{2n+3}\) converge or diverge?
  7. Does \(\sum_{m=1}^\infty\frac{1}{1+2+\dots+(m-1)+m}\) converge or diverge? (Hint: show that \( \frac{1}{1+2+\dots+(m-1)+m} = \frac{2}{(1+m)+(2+m-1)+\dots+(m-1+2)+(m+1)} \).)
  8. Does \(\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}\) converge or diverge?
  9. Does \(\sum_{n=1}^\infty\sqrt{\frac{n+1}{n^2+3}}\) converge or diverge?

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.4