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Section 6.4 Calculus 2

Taylor's Formula
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6.4 Taylor’s Formula

6.4.1 Taylor’s Formula

  • Taylor’s Formula guarantees that if \(f\) has derivatives of all orders on an open interval containing \(a\), then for every nonnegative integer \(n\) and \(x\) in that same interval, \(f(x)=\left(\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\right)+R_n(x)\) where the error term is given by \(R_n(x)=\frac{f^{(n+1)}(x_n)}{(n+1)!}(x-a)^{n+1}\) for some number \(x_n\) between \(a\) and \(x\).
  • Example Use the fact that \(e<4\) and Taylor’s Formula to estimate the value of \(\sqrt{e}\) with an error no greater than \(0.01\).
  • Example Use Taylor’s Formula to estimate the value of \(\sin(0.1)\) with an error no greater than \(0.0001\).

6.4.2 Convergence of Taylor and Maclaurin Series

  • A Taylor series converges to its generating function when \(\lim_{n\to\infty}|R_n(x)|=0\).
  • Example Prove that \(e^x=\sum_{k=0}^\infty\frac{x^k}{k!}\).
  • Example Prove that \(\cos(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}\).

Review Exercises

  1. Use the fact that \(e<3\) and Taylor’s Formula to estimate the value of \(e\) with an error no greater than \(0.001\).
  2. Use Taylor’s Formula to estimate the value of \(\cos(0.1)\) with an error no greater than \(0.0001\).
  3. Use Taylor’s Formula to estimate the value of \(\sin(1)\) with an error no greater than \(0.01\).
  4. Prove that \(\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}\).
  5. Use the fact that \(|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)\) for any \(x_n\) between \(0\) and \(x\) to prove that \(\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\).
  6. Reprove \(\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\) by using its definition \(\cosh(x)=\frac{1}{2}(e^x+e^{-x})\) along with the Maclaurin series for \(e^x\).

Solutions