Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = 1/(4 + x²), a = 0

Verified step by step guidance

1. Identify the function and the center point: here, the function is \(f(x) = \frac{1}{4 + x^{2}}\) and the center is \(a = 0\).
2. Recall the Taylor series formula 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 \(x = a\).
3. Calculate the first few derivatives of \(f(x)\) and evaluate each at \(x=0\): - Compute \(f(0)\), - Compute \(f'(0)\), - Compute \(f''(0)\), - Compute \(f'''(0)\) if needed, until you have enough terms to write the first three nonzero terms.
4. Write out the first three nonzero terms explicitly using the formula: \[f(x) \approx f(0) + f'(0) x + \frac{f''(0)}{2!} x^{2} + \cdots\] Include only the terms with nonzero coefficients.
5. Express the Taylor series in summation notation centered at \(a=0\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}\] where the derivatives \(f^{(n)}(0)\) are as calculated.

Key Concepts

Taylor Series Expansion

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

Derivatives of the Function

To find the Taylor series terms, you must compute the function's derivatives at the center point. These derivatives determine the coefficients of each term in the series. For rational functions like 1/(4 + x²), derivatives can be found using the chain and quotient rules.

Summation Notation for Series

Summation notation concisely expresses infinite series using the sigma symbol (∑). After finding the general term of the Taylor series, it can be written as a sum from n=0 to infinity, showing the pattern of coefficients and powers of (x - a). This notation simplifies representation and analysis.