Taylor series

b. Write the power series using summation notation.

f(x) = ln x, a = 3

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Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by the formula: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \] where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).

Identify the function and the center: here, \( f(x) = \ln x \) and \( a = 3 \). We need to find the derivatives of \( \ln x \) evaluated at \( x = 3 \).

Compute the first few derivatives of \( f(x) = \ln x \): - \( f(x) = \ln x \) - \( f'(x) = \frac{1}{x} \) - \( f''(x) = -\frac{1}{x^2} \) - \( f^{(3)}(x) = \frac{2}{x^3} \) - \( f^{(4)}(x) = -\frac{6}{x^4} \) Notice the pattern in the derivatives and their signs.

Evaluate each derivative at \( x = 3 \) to get \( f^{(n)}(3) \). For example, \( f'(3) = \frac{1}{3} \), \( f''(3) = -\frac{1}{9} \), and so on.

Write the Taylor series in summation notation using the derivatives evaluated at \( a = 3 \): \[ \ln x = \sum_{n=0}^{\infty} \frac{f^{(n)}(3)}{n!} (x - 3)^n \] where each \( f^{(n)}(3) \) is substituted with the corresponding derivative value found in the previous step.

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주요 개념

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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. For a function f(x) centered at a point a, the series is given by f(x) = Σ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ, where n! is factorial and f⁽ⁿ⁾(a) is the nth derivative at a.

Derivatives of the Natural Logarithm Function

The function f(x) = ln(x) has derivatives that follow a specific pattern: the first derivative is 1/x, the second is -1/x², the third is 2/x³, and so on, alternating in sign and involving factorial terms. Understanding this pattern is essential to find the coefficients of the Taylor series at a given point.

Summation Notation for Power Series

Summation notation (Σ) concisely expresses infinite sums, such as power series. Writing a Taylor series in summation form involves identifying the general term that includes the nth derivative, factorial denominator, and (x - a)ⁿ factor, allowing a compact and clear representation of the series.

Taylor seriesb. Write the power series using summation notation.f(x) = ln x, a = 3