Calc 2: Section 11-10 - Taylor and Maclaurin Series

Welcome back to Calc 2! In today's session, we continued our exploration of infinite series, specifically focusing on Taylor and Maclaurin series. These series provide a way to represent a function as an infinite sum of terms involving its derivatives at a single point. This representation allows us to approximate function values and gain deeper insights into their behavior.

Key Concepts:

  • Power Series Representation: A power series centered at $a$ is given by: $$f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + ... = \sum_{n=0}^{\infty} c_n(x-a)^n$$
  • Taylor Series: The Taylor series of a function $f(x)$ about $x=a$ is: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$
  • Maclaurin Series: A Maclaurin series is simply a Taylor series centered at $a=0$: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$$
  • Coefficient Formula: The coefficients $c_n$ in a Taylor series are determined by the derivatives of the function evaluated at the center $a$: $$c_n = \frac{f^{(n)}(a)}{n!}$$

Examples:

Let's solidify these concepts with some examples!

  1. Example 1: Finding the Maclaurin series for $f(x) = e^x$

    We know that the derivative of $e^x$ is always $e^x$. Therefore, $f^{(n)}(0) = e^0 = 1$ for all $n$. Substituting this into the Maclaurin series formula, we get: $$e^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$

    Using the ratio test, we can confirm that this series converges for all $x$ (i.e., the radius of convergence is $R = \infty$).

  2. Example 2: Finding the Maclaurin series for $f(x) = cos(x)$

    We need to find the derivatives of $cos(x)$ and evaluate them at $x=0$:

    • $f(x) = cos(x) \Rightarrow f(0) = 1$
    • $f'(x) = -sin(x) \Rightarrow f'(0) = 0$
    • $f''(x) = -cos(x) \Rightarrow f''(0) = -1$
    • $f'''(x) = sin(x) \Rightarrow f'''(0) = 0$
    • $f^{(4)}(x) = cos(x) \Rightarrow f^{(4)}(0) = 1$

    Notice the pattern! Plugging these values into the Maclaurin series, we only get even powers of $x$:

    $$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}$$
  3. Common Maclaurin Series:
    • $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ...$, $R = 1$
    • $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$, $R = \infty$
    • $sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$, $R = \infty$
    • $cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$, $R = \infty$

Keep practicing, and you'll master these powerful series! Remember to always check the radius of convergence to ensure your approximations are valid. See you in the next lecture!