Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

5.E: Convergence of the Taylor Series: A “Tayl” of Three Remainders (Exercises)

  • Page ID
    7947
  • [ "article:topic", "authorname:eboman" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Q1

    Find the Integral form, Lagrange form, and Cauchy form of the remainder for Taylor series for the following functions expanded about the given values of \(a\).

    1. \(f(x) = e^x\), \(a = 0\)
    2. \(f(x) = \sqrt{x}\), \(a = 1\)
    3. \(f(x) = (1 + x)^α\), \(a = 0\)
    4. \(f(x) = \frac{1}{x}\), \(a = 3\)
    5. \(f(x) = \ln x\), \(a = 2\)
    6. \(f(x) = \cos x\), \(a = \frac{\pi }{2}\) 

    Contributor

    • Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)