Taylor series and interval of convergence


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


f(x)=2/(1−x)³, a=0

Verified step by step guidance

1. Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a=0\) (Maclaurin series): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\] where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
2. Start by finding the function and its first few derivatives: - \(f(x) = \frac{2}{(1-x)^3}\) Calculate \(f'(x)\), \(f''(x)\), and \(f'''(x)\) using the chain rule and power rule.
3. Evaluate each derivative at \(x=0\): - Compute \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f'''(0)\) to find the coefficients for the Taylor series terms.
4. Write the first four nonzero terms of the Taylor series using the formula: \[\frac{f^{(n)}(0)}{n!} x^n\] for \(n=0,1,2,3\).
5. Express the partial sum of the Taylor series with these four terms explicitly, which approximates \(f(x)\) near \(x=0\).

Key Concepts

Taylor and Maclaurin Series

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative of the function evaluated at the center, multiplied by (x - a)^n and divided by n!.

Derivatives of the Function

To find the Taylor series terms, you must compute successive derivatives of the function at the center point. These derivatives determine the coefficients of the series. For f(x) = 2/(1−x)^3, calculating the first few derivatives at x=0 is essential to find the first four nonzero terms.

Interval of Convergence

The interval of convergence is the range of x-values for which the Taylor series converges to the function. It depends on the function's singularities and can be found using tests like the ratio test. For rational functions like 2/(1−x)^3, the interval is typically related to the distance from the center to the nearest singularity.