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

Section 5.1 Calculus 2

Sequences
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5.1 Sequences

5.1.1 Definition

  • A sequence is an infinitely long list of real numbers. For example, the sequence of positive even integers is \(\<2,4,6,8,\dots\>\).
  • Example Use your intuition to guess the next three terms of the sequences \(\<1,3,5,7,9,\dots\>\), \(\<3,-6,9,-12,15,\dots\>\), and \(\<0,1,4,9,16,\dots\>\).
  • An explicit formula \(a_n\) is a rule defining each term of the sequence, where \(n=0\) yields the first term, \(n=1\) gives the next term, and so on. The sequence generated by the formula \(a_n\) is written as \(\<a_n\>_{n=0}^\infty=\<a_0,a_1,a_2,\dots\>\).
    • Occasionally the first term of the sequence may be given by an integer different from \(0\), in which case the sequence is written like \(\<a_n\>_{n=1}^\infty\).
  • Example Write the first five terms of the sequences \(\<a_n\>_{n=0}^\infty\), \(\<b_n\>_{n=0}^\infty\), and \(\<c_n\>_{n=0}^\infty\) defined by \(a_n=4n\), \(b_n=\frac{(-1)^n}{n^2+2}\), and \(c_n=\cos(\frac{\pi}{2}n)\).
  • Example Give the term \(a_7\) for the sequence defined by the formula \(a_n=\frac{n}{2n+1}\).

5.1.2 Recursive Formulas

  • A recursive formula for a sequence defines one or more initial terms of the sequence, and then defines future terms of the sequence by using previous terms.
  • Example Write the first ten terms of the Fibonacci sequence defined by the recursive formula \(f_0=1,f_1=1,f_{n+2}=f_n+f_{n+1}\).
  • Example Write the first six terms of the factorial sequence defined by the recursive formula \(!_0=1,!_{n+1}=(n+1)!_n\).
  • The factorial sequence is commonly written in the form \(n!\) rather than \(!_n\). It has the explicit formula \(n!=1\times2\times3\times\dots\times n\).
  • Example Prove that \(a_n=\frac{3}{2^n}\) is an explicit formula for the sequence \(\<a_n\>_{n=0}^\infty\) defined recursively by \(a_0=3,a_{n+1}=\frac{a_n}{2}\).

5.1.3 Limits, Convergence, and Divergence

  • The sequence \(\<a_n\>_{n=i}^\infty\) converges to a limit \(L\) if for each \(\epsilon>0\), there exists an integer \(N\) such that \(|a_n-L|<\epsilon\) for all \(n\geq N\). This is written as \(\lim_{n\to\infty}a_n=L\) or \(a_n\to L\).
  • Example Guess the limit of the harmonic sequence
    \(\<a_n\>_{n=1}^\infty\) defined by \(a_n=\frac{1}{n}\) by writing out the first few terms.
  • Example Guess the limit of the sequence
    defined by \(g_n=\frac{2^n}{2^{n+1}}\) by writing out the first few terms.
  • A sequence diverges when it doesn’t converge to any limit.
  • Example Write a few terms of the sequence defined by the formula \(b_n=(-1)^n\frac{n+1}{n+2}\). Does it appear to be converging or diverging?

Review Exercises

  1. Use your intuition to guess the next three terms of the sequences \(\<1,5,9,13,17,\dots\>\), \(\<1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots\>\), and \(\<\frac{1}{3},-1,3,-9,27,\dots\>\).
  2. Create an explicit formula for each of the three previous sequences.
  3. Write the first five terms of the sequences \(\<a_n\>_{n=0}^\infty\), \(\<b_n\>_{n=0}^\infty\), and \(\<c_n\>_{n=0}^\infty\) defined by \(a_n=3n+2\), \(b_n=2(-\frac{1}{3})^n\), and \(c_n=\frac{n}{1+n^2}\).
  4. Write the first six terms of the sequence \(\<q_n\>_{n=0}^\infty\) defined by \(q_0=0\) and \(q_{n+1}=q_n+2n+1\).
  5. Prove that \(q_n=n^2\) is an explicit formula for the sequence defined recursively in the previous problem.
  6. Write the first six terms of the sequence \(\<b_n\>_{n=1}^\infty\) defined by \(b_1=4\) and \(b_{n+1}=\frac{b_n}{2}\).
  7. Prove that \(b_n=\frac{8}{2^n}\) is an explicit formula for the sequence defined recursively in the previous problem.
  8. Guess the limit of the alternating harmonic sequence
    \(\<b_n\>_{n=1}^\infty\) defined by \(b_n=\frac{(-1)^n}{n}\) by writing out the first few terms.
  9. Guess the limit of the geometric sequence
    \(\<g_n\>_{n=0}^\infty\) defined by \(g_n=2^{-n}\) by writing out the first few terms.
  10. Guess the limit of the sequence
    \(\<a_n\>_{n=3}^\infty\) defined by \(a_n=\frac{3n+2}{2n+1}\) by writing out the first few terms.
  11. Write a few terms of the sequence defined by the formula \(c_n=\frac{n!}{n^2+1}\). Does it appear to be converging or diverging?
  12. Write a few terms of the sequence defined by the formula \(s_n=\sin(\frac{\pi n}{3})\). Does it appear to be converging or diverging?
  13. Sketch a picture which explains why \(\lim_{n\to\infty} \sin(\pi n)=0\) as the limit of a sequence, but \(\lim_{x\to\infty}\sin(\pi x)\) does not exist as a limit of a function.
  14. What are the first five terms of the sequence \(\<r_n\>_{n=1}^\infty\) defined explicitly by \(r_n=\frac{n+2}{3+n^2}\)?
  15. What are the first five terms of the sequence \(\<w_n\>_{n=0}^\infty\) defined recursively by \(w_0=1\), \(w_1=2\), \(w_{n+2}=2w_n+w_{n+1}\)?
  16. Which of these statements seems most appropriate for describing the sequence whose initial terms are \(\<1,\frac{3}{4},\frac{5}{8},\frac{9}{16},\frac{17}{32},\dots\>\)?
    • The sequence appears to converge to \(\frac{1}{2}\).
    • The sequence appears to diverge to \(\frac{1}{2}\).
    • The sequence appears to neither converge nor diverge.

Solutions 1-7

Solutions 8-16

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.1