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Section 6.2 Calculus 2

Taylor and Maclaurin Series
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6.2 Taylor and Maclaurin Series

6.2.1 Power Series from Functions

  • Let \(f(x)\) have derivatives of all orders nearby \(a\). Then the Taylor series generated by \(f\) at \(a\) is given by \(\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k\), where \(f^{(k)}(a)\) is the \(k^{th}\) derivative of \(f\) at \(a\).
  • A Maclaurin series is a Taylor series where \(a=0\).
  • A Taylor/Maclaurin series is said to converge to its generating function if it is equal to it for all members of its domain.
  • Example Let \(f(x)=\frac{1}{1+x}\) with the domain \(-1<x<1\), and guess a formula for \(f^{(k)}(0)\) by computing its first few terms. Then show that the Maclaurin series generated by \(f\) converges to \(f\).
  • Example Let \(g(x)=\frac{2}{x}\) with the domain \(0<x<4\), and guess a formula for \(g^{(k)}(2)\) by computing its first few terms. Then show that the Taylor series generated by \(g\) at \(2\) converges to \(g\).
  • It can be shown that \(f\) defined by \(f(0)=0\) and \(f(x)=e^{-1/x^2}\) otherwise satisfies \(f^{(k)}(0)=0\), giving an example of a function which doesn’t converge to its Taylor series.
  • If a power series of the form \(\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k\) converges to \(f(x)\), then that power series is the Taylor series generated by \(f(x)\) at \(a\).

6.2.2 Maclaurin Series for \(e^x\), \(\sin x\), \(\cos x\)

  • The following Maclaurin Series can be shown to converge to their generating functions:
    • \(e^x=\sum_{k=0}^\infty\frac{x^k}{k!} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\dots\)
    • \(\cos x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!} = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots\)
    • \(\sin x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!} = x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\dots\)
  • Example Show how to generate the Maclaurin series for \(e^x\).
  • Example Show how to generate the Maclaurin series for \(\sin x\).

Review Exercises

  1. Let \(f(x)=\frac{1}{1-x}\) with the domain \(-1<x<1\), and guess a formula for \(f^{(k)}(0)\) by computing its first few terms. Then show that the Maclaurin series generated by \(f\) converges to \(f\).
  2. Let \(g(x)=\frac{3}{x}\) with the domain \(0<x<6\), and guess a formula for \(g^{(k)}(3)\) by computing its first few terms. Then show that the Taylor series generated by \(g\) at \(3\) converges to \(g\).
  3. Let \(h(x)=\frac{1}{x^2+1}\) with the domain \(-1<x<1\). It may be shown that the first few terms of \(\<h^{(k)}(0)\>_{k=0}^\infty\) are \(\<1,0,-2,0,24,0,-720,\dots\>\). Show that the Maclaurin series generated by \(h\) converges to \(h\).
  4. Show how to generate the Maclaurin series \(\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}\) for \(\cos x\).
  5. Find the Maclaurin series for \(\sinh x\).
  6. Find the Maclaurin series for \(\cosh x\).
  7. Find the Maclaurin series for \(e^{-x}\).
  8. Find the Maclaurin series for \(x^3+3x-7\).

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.8