Module 4: Power Series - Summary Notes

Power Series

  • A power series in xx is an infinite series with non-negative integral powers of xx: c<em>0+c</em>1x+c<em>2x2+⋯=∑</em>n=0∞cnxnc<em>0 + c</em>1x + c<em>2x^2 + \cdots = \sum</em>{n=0}^{\infty} c_nx^n.
  • A more general form is a power series in (x−a)(x - a): c<em>0+c</em>1(x−a)+c<em>2(x−a)2+⋯+c</em>n(x−a)n+⋯=∑<em>n=0∞c</em>n(x−a)nc<em>0 + c</em>1(x - a) + c<em>2(x - a)^2 + \cdots + c</em>n(x - a)^n + \cdots = \sum<em>{n=0}^{\infty} c</em>n(x - a)^n.

Taylor Series

  • Represents a function f(x)f(x) as a power series: f(x)=∑<em>n=0∞c</em>n(x−a)n=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum<em>{n=0}^{\infty} c</em>n(x - a)^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n for |x - a| < r.
  • Coefficients c<em>nc<em>n are related to derivatives of f(x)f(x) at aa: c</em>n=f(n)(a)n!c</em>n = \frac{f^{(n)}(a)}{n!}.

Maclaurin Series

  • A Maclaurin series is a Taylor series with a=0a = 0:
    f(x)=∑n=0∞f(n)(0)n!xn=f(0)+f′(0)x+f′′(0)2!x2+f′′′(0)3!x3+⋯f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots.

Radius of Convergence

  • The interval of convergence is the set of all real numbers xx for which a power series converges.
  • A power series in (x−a)(x - a) may converge on a finite interval centered at aa, on the infinite interval (−∞,∞)(-\infty, \infty), or at the single point x=ax = a.
  • The radius of convergence is rr, ∞\infty, or 00, respectively.

Manipulation of Power Series

  • Addition and Subtraction: Power series can be added or subtracted term-by-term within their common interval of convergence.
    f(x)±g(x)=∑<em>n=0∞(c</em>n±d<em>n)xnf(x) \pm g(x) = \sum<em>{n=0}^{\infty} (c</em>n \pm d<em>n)x^n for ∣x∣<min⁡(r</em>1,r2)|x| < \min(r</em>1, r_2).
  • Composition: Substitute a function g(x)g(x) into a power series f(x)f(x).
    f(g(x))=∑<em>n=0∞c</em>n[g(x)]nf(g(x)) = \sum<em>{n=0}^{\infty} c</em>n [g(x)]^n.
  • Multiplication: Multiply power series term by term.
    f(x)⋅g(x)=(c<em>0+c</em>1x+c<em>2x2+⋯ )(d</em>0+d<em>1x+d</em>2x2+⋯ )=c<em>0d</em>0+(c<em>0d</em>1+c<em>1d</em>0)x+(c<em>2d</em>0+c<em>1d</em>1+c<em>0d</em>2)x2+⋯f(x) \cdot g(x) = (c<em>0 + c</em>1x + c<em>2x^2 + \cdots)(d</em>0 + d<em>1x + d</em>2x^2 + \cdots) = c<em>0d</em>0 + (c<em>0d</em>1 + c<em>1d</em>0)x + (c<em>2d</em>0 + c<em>1d</em>1 + c<em>0d</em>2)x^2 + \cdots.
  • Division: Divide power series using long division or by equating coefficients.
    f(x)g(x)=q<em>0+q</em>1x+q<em>2x2+⋯\frac{f(x)}{g(x)} = q<em>0 + q</em>1x + q<em>2x^2 + \cdots. The resulting power series may not converge for ∣x∣<min⁡(r</em>1,r2)|x| < \min(r</em>1, r_2)