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Section 6.1 Calculus 2

Power Series
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6.1 Power Series

6.1.1 Definition

  • A power series is a function defined by \(f(x)=\sum_{n=0}^\infty c_n(x-a)^n= c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+\dots\) for a coefficient sequence \(\<c_n\>_{n=0}^\infty\) and center \(a\).
  • Example Expand the first four terms of the power series \(\sum_{m=0}^\infty (2m+1)x^m\).
  • Example Expand the first four terms of the power series \(\sum_{k=2}^\infty \frac{x^{2k}}{k!}\).
  • The geometric series formula \(\sum_{n=0}^\infty ar^n\) allows us to simplify certain series where \(|r|<1\).
  • Example Simplify \(f(x)=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\dots\) with domain \(|x|<1\).
  • Example Simplify \(p(x)=\sum_{j=1}^\infty (\frac{x}{3})^{2j}= \frac{x^2}{9}+\frac{x^4}{81}+\frac{x^6}{729}+\dots\) with domain \(|x|<3\).

6.1.2 Domains of Power Series

  • The domain of a power series may be determined by applying the Root or Ratio Test to determine for which \(x\) values the series converges. On the endpoints where these tests are inconclusive, other techniques must be used.
  • Example Find the domain of \(f(x)=\sum_{n=1}^\infty\frac{x^n}{n}=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots\).
  • Example Find the domain of \(h(x)=\sum_{n=0}^\infty\frac{(3-2x)^n}{n!}= 1+(3-2x)+\frac{(3-2x)^2}{2}+\frac{(3-2x)^3}{6}+\dots\).
  • Example Find the domain of \(g(x)=\sum_{k=2}^\infty\frac{(3x)^n}{n\ln n}= \frac{9x^2}{2\ln2}+\frac{27x^3}{3\ln3}+\frac{81x^4}{4\ln4}+\dots\).

Review Exercises

  1. Expand the first four terms of the power series \(\sum_{m=0}^\infty 3^{m+1}x^m\).
  2. Expand the first four terms of the power series \(\sum_{k=1}^\infty \frac{(-x)^k}{k+1}\).
  3. Expand the first four terms of the power series \(\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}\).
  4. Simplify \(q(x)=\sum_{n=1}^\infty (1-x)^n=(1-x)+(1-x)^2+(1-x)^3+\dots\) with domain \(|1-x|<1\).
  5. Simplify \(g(x)=\sum_{j=0}^\infty (2x)^{2j+1}= 2x+8x^3+32x^5+128x^7+\dots\) with domain \(|x|<\frac{1}{2}\).
  6. Find the domain of \(z(x)=\sum_{n=2}^\infty(-1)^n\frac{x^n}{2n}= \frac{x^2}{4}-\frac{x^3}{6}+\frac{x^4}{8}-\frac{x^5}{10}+\dots\).
  7. Find the domain of \(f(x)=\sum_{i=0}^\infty\frac{(3x)^i}{(2i)!}= 1+\frac{3}{2}x+\frac{9}{24}x^2+\dots\).
  8. Find the domain of \(h(x)=\sum_{k=0}^\infty\frac{(x-2)^k}{k^2+1}= 1+\frac{x-2}{2}+\frac{(x-2)^2}{5}+\frac{(x-2)^3}{10}+\dots\).
  9. Find the domain of \(g(x)=\sum_{m=3}^\infty(\frac{1}{m}-\frac{1}{m+1})x^m\).
  10. Expand the first four terms of the power series \(\sum_{k=0}^\infty \frac{(-x)^{2k+1}}{(2k)!}\).
  11. Simplify \(f(x)=\sum_{n=1}^\infty (-x)^{n-1}=1-x+x^2-x^3+\dots\) with domain \(|x|<1\).
  12. Find the domain of \(f(x)=\sum_{m=2}^\infty\frac{(-2x)^m}{m}= \frac{4x^2}{2}-\frac{8x^3}{3}+\frac{16x^4}{4}-\frac{32x^5}{5}+\dots\).

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 9.7