Calc 2 - 10/29

Taylor and Maclaurin Series

  • Taylor Series: Centered at any point $a$.
  • Maclaurin Series: Special case of Taylor Series where the center is at $0$.
  • Expansion based on the $n$th derivative of the function evaluated at the center.

Exponential Function

  • The series representation of $e^x$ is given by:
    ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
  • Valid for all $x$ in (−∞,∞)(-\infty, \infty).

Simplification and Derivatives

  • Simplifying 2n+1(2n+1)!\frac{2n + 1}{(2n + 1)!} involves expressing the factorial in the denominator.
  • For derivatives of functions like $x^{2n+1}$, use the power rule:
    • Derivative: ddx(x2n+1)=(2n+1)x2n\frac{d}{dx}(x^{2n+1}) = (2n + 1)x^{2n}.

Integration of Series

  • Integral of e4te^{4t} requires treating $x$ as a constant during integration.
  • Result: ∫<em>02xe4tdt=∑</em>n=0∞4nn!∫tndt\int<em>0^{2x} e^{4t} dt = \sum</em>{n=0}^{\infty} \frac{4^n}{n!} \int t^n dt.

Maclaurin Series for Sine Function

  • The Maclaurin series for sin⁡(x)\sin(x):
    • Function and derivatives yield patterns: f(0)=0,f′(0)=1,f′′(0)=0,f′′′(0)=−1f(0)=0, f'(0)=1, f''(0)=0, f'''(0)=-1 (alternating pattern).
    • Series representation:
      sin⁡x=∑n=0∞(−1)n(2n+1)!x2n+1\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!}x^{2n + 1}.

Radius of Convergence

  • Found via the ratio test:
    • R=lim⁡<em>n→∞∣a</em>n+1an∣R = \lim<em>{n \to \infty} \left| \frac{a</em>{n+1}}{a_n} \right|
    • For sine and cosine series: Radius of convergence is (−∞,∞)(-\infty, \infty).

Cosine Function through Differentiation

  • Maclaurin series of cos⁡(x)\cos(x) derived from sin⁡(x)\sin(x) by differentiation:
    • cos⁡(x)=∑n=0∞(−1)n(2n)!x2n\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}.

Binomial Series

  • Expansion for functions in the form (1+x)k(1 + x)^k:
    • Coefficients given by:
      C(k,n)=k(k−1)(k−2)…(k−n+1)n!C(k, n) = \frac{k(k-1)(k-2)…(k-n+1)}{n!}.
  • Valid for the series when |x| < 1.

Example Transformation for Binomial Series

  • To rewrite 8−x43\sqrt[3]{8 - x^4}: Change to 81/3(1−x48)1/38^{1/3}(1 - \frac{x^4}{8})^{1/3} and apply binomial expansion.