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) = 1/x, a = 1

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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 \(x = a\).

Identify the function and center: here, \(f(x) = \frac{1}{x}\) and \(a = 1\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=1\).

Compute the first four derivatives of \(f(x)\): - \(f(x) = x^{-1}\) - \(f'(x) = -x^{-2}\) - \(f''(x) = 2x^{-3}\) - \(f^{(3)}(x) = -6x^{-4}\) - \(f^{(4)}(x) = 24x^{-5}\)

Evaluate each derivative at \(x = 1\): - \(f(1) = 1\) - \(f'(1) = -1\) - \(f''(1) = 2\) - \(f^{(3)}(1) = -6\) - \(f^{(4)}(1) = 24\)

Write the first four nonzero terms of the Taylor series using the formula: \[f(x) \approx f(1) + \frac{f'(1)}{1!}(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f^{(3)}(1)}{3!}(x-1)^3,\] substituting the values found for each derivative.

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주요 개념

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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, called the center. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!. This series approximates the function near the center point.

Derivatives of the Function

To construct the Taylor series, you need to compute successive derivatives of the function at the center point. For f(x) = 1/x, derivatives involve powers of x with alternating signs. Evaluating these derivatives at a = 1 provides the coefficients for the series terms.

Constructing the Series Terms

Each term of the Taylor series is formed by dividing the nth derivative at the center by n! and multiplying by (x - a)^n. Identifying the first four nonzero terms requires calculating derivatives up to the third order and substituting into this formula to write the polynomial approximation.

Taylor seriesa. 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) = 1/x, a = 1