Taylor series


a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x) = ln x, a = 3

Verified step by step guidance

1. Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
2. Identify the function and center: here, \(f(x) = \ln x\) and \(a = 3\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=3\).
3. Calculate the first four derivatives of \(f(x) = \ln x\): - \(f(x) = \ln x\) - \(f'(x) = \frac{1}{x}\) - \(f''(x) = -\frac{1}{x^2}\) - \(f^{(3)}(x) = \frac{2}{x^3}\) - \(f^{(4)}(x) = -\frac{6}{x^4}\)
4. Evaluate each derivative at \(x = 3\): - \(f(3) = \ln 3\) - \(f'(3) = \frac{1}{3}\) - \(f''(3) = -\frac{1}{9}\) - \(f^{(3)}(3) = \frac{2}{27}\) - \(f^{(4)}(3) = -\frac{6}{81} = -\frac{2}{27}\)
5. Write the first four nonzero terms of the Taylor series using the formula: \[f(x) \approx f(3) + f'(3)(x-3) + \frac{f''(3)}{2!}(x-3)^2 + \frac{f^{(3)}(3)}{3!}(x-3)^3 + \frac{f^{(4)}(3)}{4!}(x-3)^4,\] substituting the values found for each derivative.

Key Concepts

Taylor Series Definition

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. Each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!. This series approximates the function near the point a.

Derivatives of the Natural Logarithm Function

To find the Taylor series of f(x) = ln(x), you need to compute its derivatives at x = a. The first derivative is 1/x, and higher derivatives follow a pattern involving factorials and powers of x. Understanding these derivatives is essential to form the series terms.

Constructing the First Four Nonzero Terms

After finding the derivatives at a, substitute them into the Taylor series formula to write out the first four nonzero terms. This involves careful calculation of each term’s coefficient and power of (x - a), ensuring the series accurately approximates ln(x) near x = 3.