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) = (1 + x²)⁻¹, a = 0

Verified step by step guidance

1. Recall that the Maclaurin series is a Taylor series centered at \(a = 0\), and is given by the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\] where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
2. Start by finding the function and its derivatives at \(x=0\). The function is \(f(x) = (1 + x^2)^{-1}\). Calculate the first few derivatives \(f'(x)\), \(f''(x)\), \(f^{(3)}(x)\), and \(f^{(4)}(x)\).
3. Evaluate each derivative at \(x=0\) to find \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f^{(3)}(0)\). These values will be used as coefficients in the series.
4. Write out the first four nonzero terms of the Maclaurin series using the formula: \[\frac{f(0)}{0!} x^0 + \frac{f'(0)}{1!} x^1 + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots\] Include only the terms where the coefficient is nonzero.
5. Simplify each term to express the series clearly, and verify the pattern of coefficients to confirm the first four nonzero terms of the Taylor series centered at 0.

Key Concepts

Taylor and Maclaurin Series

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

Derivatives and Term Calculation

To find the Taylor series terms, compute successive derivatives of the function at the center point. Each derivative provides the coefficient for the corresponding term in the series. Identifying the first four nonzero terms requires careful differentiation and evaluation at a = 0.

Interval of Convergence

The interval of convergence is the range of x-values for which the Taylor series converges to the function. It is determined by testing the series' convergence, often using ratio or root tests. Understanding this interval ensures the series accurately represents the function within that domain.