Taylor Series: Concepts and Applications

Power Series

Taylor Series

The Taylor series is a powerful tool in calculus for representing functions as infinite sums of their derivatives evaluated at a single point. It is a specific type of power series that approximates functions near a given point, often used for analysis, computation, and solving differential equations.

• Definition: The Taylor series of a function f(x) centered at a is given by: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$

• Key Terms:

  • Power series: An infinite sum of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$.

  • Maclaurin series: A Taylor series centered at a = 0.

• Convergence: The Taylor series converges to f(x) within the radius of convergence, which depends on the function and the point a.

• Applications:

  • Approximating functions near a point

  • Solving differential equations

  • Calculating limits and integrals

Examples of Taylor Series

• Maclaurin Series for $e^x$: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

• Maclaurin Series for $\sin x$: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$

• Maclaurin Series for $\cos x$: $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$

Properties and Comparison

• Accuracy: The more terms included, the closer the approximation to the actual function within the radius of convergence.

• Remainder (Error): The error in approximating f(x) by its Taylor polynomial of degree n is given by: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1}$ where c is between a and x.

Table: Common Taylor Series (Maclaurin Series)

Function	Maclaurin Series	Interval of Convergence
$e^x$	$\sum_{n=0}^{\infty} \frac{x^n}{n!}$	All real numbers
$\sin x$	$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$	All real numbers
$\cos x$	$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$	All real numbers
$\ln(1+x)$	$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n$	$-1 < x \leq 1$

Example: To approximate $e^{0.1}$ using the first three terms of its Maclaurin series: $e^{0.1} \approx 1 + 0.1 + \frac{(0.1)^2}{2} = 1 + 0.1 + 0.005 = 1.105$

Additional info: The Taylor series is a central concept in Chapter 11 (Power Series) and is foundational for advanced calculus topics, including numerical analysis and mathematical modeling.