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Section 6.3 Calculus 2

Manipulating Power Series
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6.3 Manipulating Power Series

6.3.1 Multiplication and Composition with Power Series

  • Let \(p(x)\) be a polynomial. If \(\sum_{n=0}^\infty c_n(x-a)^n\) is a power series converging to \(f(x)\), then \(\sum_{n=0}^\infty p(x)c_n(x-a)^n\) is a power series converging to \(p(x)f(x)\) with the same domain of convergence.
  • Example. Find the Maclaurin series for \(x^3\cos(x)\).
  • Let \(p(x)\) be a polynomial. If \(\sum_{n=0}^\infty c_n(x-a)^n\) is a power series converging to \(f(x)\), then \(\sum_{n=0}^\infty c_n(p(x)-a)^n\) is a power series converging to \(f(p(x))\) with an appropriate domain of convergence.
  • Example. Find the Maclaurin series for \(\exp(x^2)\).
  • Example. Find the Maclaurin series for \(x\sin(2x)\).

6.3.2 Differentiating and Integrating Power Series

  • If \(\sum_{n=0}^\infty c_n(x-a)^n\) is a power series converging to \(f(x)\) on an open interval, then \(\sum_{n=0}^\infty c_n n(x-a)^{n-1}\) is a power series converging to \(f’(x)\) with the same domain of convergence.
  • Example. Prove that \(\frac{d}{dx}[e^x]=e^x\).
  • Example. Find the Maclaurin series for \(\frac{1}{(1-x)^2}\) where \(|x|<1\).
  • If \(\sum_{n=0}^\infty c_n(x-a)^n\) is a power series converging to \(f(x)\) on an open interval, then \(\sum_{n=0}^\infty c_n \frac{(x-a)^{n+1}}{n+1} + C\) is a power series converging to \(\int f(x)\,dx\) with the same domain of convergence.
  • Example. Find a power series converging to \(\ln(x)\) where \(0<x<2\).

Review Exercises

  1. Find a power series converging to \(\frac{x^2}{1-x}\) for \(-1<x<1\).
  2. Find a power series converging to \(x^3\cos(x)\) for all \(x\).
  3. Find the Taylor series generated by \((x-\pi/2)\sin(x)\) at \(\pi/2\).
  4. Find the Maclaurin series generated by \(\sin(x^3)\).
  5. Find a power series converging to \(\frac{2}{2-x}\) for \(-2<x<2\). (Hint: start with \(\frac{1}{1-x}=\frac{2}{2-2x}\).)
  6. Find the Maclaurin series generated by \(3x^2\cos(x^3)\).
  7. Find a power series converging to \(\tan^\leftarrow(x)\) for \(-1<x<1\). (Hint: use \(\frac{1}{1-(-x^2)}\).)

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.8, 9.9