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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.R.43
Chapter 11, Problem 11.R.43

Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + x)^(1/3)"

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Recall that the Maclaurin series is the Taylor series expansion of a function about \(x = 0\). It is given by the formula: \[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots\]
Identify the function: \(f(x) = (1 + x)^{\frac{1}{3}}\). We will need to find the first and second derivatives of \(f(x)\) to get the first three terms.
Calculate the first derivative using the power rule for fractional exponents: \[f'(x) = \frac{1}{3}(1 + x)^{-\frac{2}{3}}\]
Calculate the second derivative by differentiating \(f'(x)\) again: \[f''(x) = \frac{1}{3} \times \left(-\frac{2}{3}\right)(1 + x)^{-\frac{5}{3}} = -\frac{2}{9}(1 + x)^{-\frac{5}{3}}\]
Evaluate \(f(0)\), \(f'(0)\), and \(f''(0)\) by substituting \(x = 0\) into each expression, then write the first three terms of the Maclaurin series as: \[f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of its derivatives evaluated at zero, multiplied by powers of x. This series helps approximate functions near zero using polynomials.
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Binomial Series Expansion

The binomial series generalizes the binomial theorem to any real exponent, allowing expansion of expressions like (1 + x)^r. It uses coefficients derived from generalized binomial coefficients, which involve factorials or the Gamma function for non-integer powers.
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Derivatives of Power Functions

To find terms in the Maclaurin series, you compute derivatives of the function at zero. For functions like (1 + x)^(1/3), derivatives involve applying the power rule repeatedly, which helps determine coefficients for each term in the series.
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