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Section 5.4 Calculus 2

Geometric and Alternating Series
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5.4 Geometric and Alternating Series

5.4.1 Geometric Series Test

  • The geometric series \(\sum_{n=0}^\infty ar^n\) converges to \(\frac{a}{1-r}\) when \(|r|<1\), and diverges when \(|r|\geq 1\).
  • Example Compute \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots\).
  • Example Does \(\sum_{k=0}^\infty\frac{2}{3^{k+1}}\) converge or diverge? If it converges, what is its value?
  • Example Does \(\sum_{k=0}^\infty\frac{2}{(1/3)^{k+1}}\) converge or diverge? If it converges, what is its value?

(TODO record)

  • Example Prove that \(0.\overline4=0.444\dots\) equals \(\frac{4}{9}\) by expressing the decimal expression as a geometric series.

5.4.2 Alternating Series Test

  • Let \(\<a_n\>_{n=0}^\infty\) be a monotonic sequence of positive terms. Then \(\sum_{n=0}^\infty(-1)^n a_n=a_0-a_1+a_2-a_3+\dots\) is called an alternating series.
  • An alternating series \(\sum_{n=0}^\infty(-1)^n a_n\) converges when \(\lim_{n\to\infty} a_n = 0\), and diverges otherwise.
  • This test cannot be used to tell what value the alternating series converges to.
  • This test also holds for \(\sum_{n=N}^\infty(-1)^{n\pm k} a_n\), as long as the terms \(a_n\) are positive and monotonic.
  • Example Show that the alternating harmonic series \(\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n} =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots\) converges.
  • Example Does the series \(\sum_{n=0}^\infty\cos(n\pi)(3+\frac{1}{n^2+1})\) converge or diverge?
  • Example Does the series \(\sum_{m=2}^\infty(-1)^m\frac{m}{m^{3/2} +3}\) converge or diverge?

Review Exercises

  1. Compute \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots\).
  2. Prove that \(0.\overline3=0.333\dots\) equals \(\frac{1}{3}\) by expressing the decimal expression as a geometric series.
  3. Write \(5.\overline{27}=5.272727\dots\) as a fraction of integers.
  4. Does \(\sum_{n=0}^\infty\frac{6}{3^{n+2}}\) converge or diverge? If it converges, what is its value?
  5. Does \(\sum_{m=0}^\infty 3(-1)^m\) converge or diverge? If it converges, what is its value?
  6. Prove \(\sum_{n=1}^\infty\frac{1}{3^n}=\frac{1}{2}\) by mimicking the proof of the Geometric Series formula.
  7. Does \(\sum_{n=0}^\infty(-1)^{n+1}\frac{4}{n^2+3}\) converge or diverge?
  8. Does \(\sum_{i=6}^\infty(-1)^i\frac{i}{\sqrt{i^3-7}}\) converge or diverge?
  9. Does \(\sum_{m=2}^\infty(-\frac{3}{5})^m\) converge or diverge?
  10. Does \(\sum_{k=2}^\infty(-\frac{5}{3})^k\) converge or diverge?
  11. Does \(\sum_{n=13}^\infty(-1)^n\frac{1}{n\ln n}\) converge or diverge?

Solutions 1-6

Solutions 7-11

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.2, 9.6