Question

Key Idea 8.8 .1 gives the \(n^{\text {th }}\) term of the Taylor series of common functions. In Exercises \(3-6,\) verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying a pattern. $$f(x)=e^{x} ; \quad c=0$$

Short answer

The Taylor series for \( e^x \) at \( c = 0 \) is \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).

Step-by-step solution

Recall the Formula for Taylor Series

The Taylor series for a function \( f(x) \) about a point \( c \) is given by \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n \]. Here, we need to find the series for \( f(x) = e^x \), centered at \( c = 0 \).

Calculate Derivatives of the Function

For \( f(x) = e^x \), all derivatives \( f^{(n)}(x) \) are also \( e^x \). Evaluating at \( c = 0 \), we have \( f^{(n)}(0) = e^0 = 1 \) for all \( n \).

Substitute into Taylor Series Formula

Substituting the derivatives into the Taylor series formula, we get: \[ f(x) = \sum_{n=0}^{\infty} \frac{1}{n!}x^n \].

Write Out the First Few Terms

Writing out the first few terms of the Taylor series: \[ f(x) = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \].

Identify the Pattern

The terms of the series are \( \frac{x^n}{n!} \), indicating a clear pattern for the formula given the Key Idea 8.8 .1: each term is derived as \( \frac{x^n}{n!} \) for each \( n \).

Key concepts

Derivatives

In calculus, derivatives represent the rate at which a function is changing at any given point. They're essentially the slopes of the tangent lines to the function's graph. Understanding derivatives is crucial when exploring Taylor series because each term in the series involves a derivative of the function. Let's break this down further:

• The first derivative, denoted as \( f'(x) \), tells us the slope of the tangent at a point \( x \).
• The second derivative, \( f''(x) \), describes the curvature of the function.
• Higher-order derivatives, such as \( f^{(n)}(x) \), give us more detailed information about the function's behavior.

For the function \( f(x) = e^x \), an interesting property arises: every derivative of \( e^x \) returns \( e^x \) itself. So, at \( x = 0 \), all derivatives of \( e^x \) are equal to 1:

• \( f'(0) = e^0 = 1 \)
• \( f''(0) = e^0 = 1 \), and so on.

This constant derivative value simplifies the computation of Taylor series coefficients.

Power Series

A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) are coefficients, \( x \) is a variable, and \( c \) is a constant. Power series are used to represent functions as sums of infinitely many terms:

• Each term in the series is a power of \( x - c \).
• The coefficient \( a_n \) often involves function derivatives.

In the context of the Taylor series for \( e^x \), we rewrite the function as a power series centered at \( c = 0 \):\[f(x) = \sum_{n=0}^{\infty} \frac{1}{n!} x^n\]Here, \( \frac{1}{n!} \) represents the coefficient \( a_n \), and each subsequent term involves a higher power of \( x \). This form is particularly useful because it allows us to approximate more complicated functions by simple polynomials. You can visualize it as the function becoming a sum of infinitely many polynomial terms.

Center of Expansion

The center of expansion, \( c \), in a Taylor series is the point around which we expand our function. It's where we evaluate the derivatives used in the series. If \( f(x) \) is expanded around \( c = 0 \), it's known as a Maclaurin series, which is a special case of the Taylor series.

• Choosing \( c = 0 \) often simplifies calculations, as seen in our example for \( e^x \).
• The series represents \( f(x) \) in terms of powers of \( (x-0) \), so \( x^n \).

The choice of a center affects how well the Taylor series approximates the function. Near the center, the approximation is more accurate. Moving away from the center, the series could diverge or be less accurate, depending on the function and the point chosen.For functions like \( e^x \), which behave nicely, choosing \( c = 0 \) offers a uniform approximation on the entire real line.This understanding is crucial when applying Taylor series in solving real-world problems, as the center of expansion significantly influences the series' usefulness and accuracy.