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Section 5.6 Calculus 2

Ratio and Root Tests
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5.6 Ratio and Root Tests

5.6.1 Ratio Test

  • The Ratio Test states that the series \(\sum_{n=N}^\infty a_n\) converges when \(\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1\) and diverges when \(\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|>1\).
  • Example Show that \(\sum_{n=0}^\infty\frac{3^n+1}{4^n}\) converges using the Ratio Test. Then give its value.
  • Example Does \(\sum_{k=3}^\infty\frac{(2k)!}{3(k!)^2}\) converge or diverge?
  • Another test must be used when \(\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1\).
  • Example Show that the divergent series \(\sum_{n=1}^\infty\frac{1}{n}\) and the convergent series \(\sum_{n=1}^\infty\frac{1}{n^2}\) both satisfy \(\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1\).

5.6.2 Root Test

  • The Root Test states that the series \(\sum_{n=N}^\infty a_n\) converges when \(\lim_{n\to\infty}\sqrt[n]{|a_n|}<1\) and diverges when \(\lim_{n\to\infty}\sqrt[n]{|a_n|}>1\).
  • Example Show that \(\sum_{n=0}^\infty\frac{5^n}{2^{3n}}\) converges using the Root Test. Then give its value.
  • Example Does \(\sum_{m=3}^\infty\frac{m^{10}}{(-3)^m}\) converge or diverge?
  • Another test must be used when \(\lim_{n\to\infty}\sqrt[n]{|a_n|}=1\).

Review Exercises

  1. Does \(\sum_{k=1}^\infty\frac{k^2+4}{(k+2)!}\) converge or diverge?
  2. Does \(\sum_{n=0}^\infty\frac{(2n)!}{n+3}\) converge or diverge?
  3. Does \(\sum_{m=2}^\infty\frac{5^m}{m!}\) converge or diverge?
  4. Does \(\sum_{n=0}^\infty(-1)^n\frac{n!}{2^n(n+2)!}\) converge or diverge?
  5. Does \(\sum_{p=0}^\infty\frac{3^p}{(p+7)^p}\) converge or diverge?
  6. Does \(\sum_{n=9}^\infty(1+\frac{2}{n})^{n^2}\) converge or diverge? (Hint: \(e^x=\lim_{n\to\infty}(1+\frac{x}{n})^n\).)
  7. Does \(\sum_{j=3}^\infty(-3)^j\frac{1}{j4^j}\) converge or diverge?
  8. Does \(\sum_{n=1}^\infty\left(\frac{1-4n^2}{(n+1)(3n+1)}\right)^{n+3}\) converge or diverge?
  9. Does \(\sum_{m=4}^\infty(-1)^{m+1}\frac{me^{-m}}{(2m+1)\ln(m+1)}\) converge or diverge?
  10. Does \(\sum_{n=1}^\infty\frac{(n-1)!}{10^n}\) converge or diverge?
  11. Does \(\sum_{k=3}^\infty(1-\frac{1}{k})^{k^2}\) converge or diverge
  12. Does \(\sum_{m=2}^\infty\frac{1}{m^2}\) converge or diverge?

Solutions

Textbook Reference

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.5