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:
  $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• Valid for all $x$ in $(-\infty, \infty)$.

Simplification and Derivatives

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

Integration of Series

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

Maclaurin Series for Sine Function

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

Radius of Convergence

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

Cosine Function through Differentiation

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

Binomial Series

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

Example Transformation for Binomial Series

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