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Section 5.3 Calculus 2

Series
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5.3 Series

5.3.1 Series as Partial Sum Sequences

  • For a given sequence \(\<a_n\>_{n=0}^\infty\), its partial sum sequence \(\<s_n\>_{n=0}^\infty\) is defined explicitly by \(s_n=\sum_{i=0}^n a_i=a_0+a_1+\dots+a_n\), and defined recursively by \(s_0=a_0\) and \(s_{n+1}=s_n+a_{n+1}\).
  • Example Write out the first few terms of the partial sum sequence for \(\<1,2,3,4,5,\dots\>\).
  • Example Write out the first few terms of the partial sum sequence for \(\<b_i\>_{i=1}^\infty\) where \(b_i=\frac{6}{i}\).
  • The series \(\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots\) represents the sum of the infinite sequence \(\<a_n\>_{n=0}^\infty\). If its partial sum sequence converges to \(L\), then we say that its series converges to \(L\) and the value of the series is \(L\) (written \(\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots=L\)). Otherwise, we say the series diverges.

5.3.2 Finding Limits of Partial Sum Sequences

  • A telescoping series is a series whose partial sum sequence allows for canceling.
  • Example Show that \(\sum_{n=1}^\infty(\frac{1}{n}-\frac{1}{n+1})\) converges to \(1\) by evaluating the limit of its partial sum sequence.
  • Example Does \(\sum_{n=0}^\infty\frac{2}{n^2+3n+2}\) converge or diverge?

(TODO record video)

  • The geometric series defined for real numbers \(a,r\) is \(\sum_{n=0}^\infty ar^n=a+ar+ar^2+ar^3+\dots\).
  • The limit of a geometric partial sum sequence \(s_n=\sum_{k=0}^\infty ar^k\) may be computed by first finding \(s_n-rs_n=a-ar^{n+1}\) and then dividing by \(r\). This limit only exists when \(|r|<1\).
  • Example Find the value of \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots\).
  • Example Does \(\sum_{k=1}^\infty \frac{4^n}{3^{n+2}}\) converge or diverge?

5.3.3 Divergent Series

  • The Series Divergence Test: If a sequence fails to converge to \(0\), then its series diverges.
  • Example Does \(\sum_{k=0}^\infty\frac{k^2+3}{2k^2+k+5}\) converge or diverge? If it converges, what is its value?
  • This does NOT mean that if a sequence converges, then its series converges.
  • The harmonic sequence \(\<\frac{1}{n}\>_{n=1}^\infty\) converges to \(0\), but its series \(\sum_{n=1}^\infty\frac{1}{n}\) diverges.

5.3.4 Arithmetic Rules and Reindexing

  • Because a series is a limit, it follows the same rules as limits do.
  • Example Evaluate the convergent series \(\sum_{i=0}^\infty\frac{1+\frac{2^{i+2}}{i+1}-\frac{2^{i+2}}{i+2}}{2^i}\).
  • The starting index for a series may be adjusted by offsetting the index for its sequence in the opposite direction.
  • Example Does \(\sum_{m=-1}^\infty\frac{1}{m+2}\) converge or diverge? If it converges, what is its value?

Review Exercises

  1. Write out the first four terms of the partial sum sequence for \(\<1,-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\dots\>\).
  2. Write out the first four terms of the partial sum sequence for \(\<0.3,0.03,0.003,0.0003,\dots\>\).
  3. Does \(\sum_{m=2}^\infty(\frac{3}{2m}-\frac{3}{2m+2})\) converge or diverge? If it converges, what is its value?
  4. Does \(\sum_{j=2}^\infty\frac{6}{4j^2+4j}\) converge or diverge? If it converges, what is its value?
  5. Compute \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots\).
  6. Prove that \(0.\overline3=0.333\dots\) equals \(\frac{1}{3}\) by expressing the decimal expression as a geometric series.
  7. Write \(0.\overline{27}=0.272727\dots\) as a fraction of integers.
  8. Does \(\sum_{n=0}^\infty\frac{6}{3^{n+2}}\) converge or diverge? If it converges, what is its value?
  9. Does \(\sum_{m=0}^\infty 3(-1)^m\) converge or diverge? If it converges, what is its value?
  10. Does \(\sum_{i=1}^\infty \frac{i+\sin i}{2i}\) converge or diverge? If it converges, what is its value?
  11. Suppose \(\sum_{n=0}^\infty a_n=3\) and \(\sum_{n=0}^\infty b_n=4\). Evaluate \(\sum_{n=0}^\infty(3a_n-2b_n)\).
  12. Does \(\sum_{k=2}^\infty 4(\frac{2}{3})^k\) converge or diverge? If it converges, what is its value?
  13. Prove \(\sum_{n=1}^\infty\frac{1}{3^n}=\frac{1}{2}\) using the proof of the Geometric Series formula (not the formula itself).
  14. Does \(\sum_{n=3}^\infty\left(\frac{6}{n}-\frac{6}{n+1}\right)\) converge or diverge? If it converges, what is its value?
  15. Does \(\sum_{i=0}^\infty\frac{(-3)^i}{2}\) converge or diverge? If it converges, what is its value?
  16. Does \(\sum_{n=1}^\infty\frac{1}{4^n}\) converge or diverge? If it converges, what is its value?

Solutions 1-7

Solutions 8-16

Textbook References

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