Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4

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Identify the function given: \(f(x) = \left(1 - \frac{x}{3}\right)^4\) and recognize that it is a binomial expression raised to a power.

Recall the Binomial Theorem for expanding \((1 + u)^n\(, which states: \[(1 + u)^n = \sum_{k=0}^{n} \binom{n}{k} u^k,\] where \)\binom{n}{k}\) is the binomial coefficient.

In this problem, set \(u = -\frac{x}{3}\) and \(n = 4\). Write the expansion as: \[\left(1 - \frac{x}{3}\right)^4 = \sum_{k=0}^4 \binom{4}{k} \left(-\frac{x}{3}\right)^k.\]

Calculate each term of the sum for \(k = 0, 1, 2, 3, 4\) by evaluating the binomial coefficients and powers of \(u\), and simplify each term to express it as a polynomial in \(x\).

Write out the first four nonzero terms explicitly from the expansion, which will form the first four nonzero terms of the Taylor series centered at \(x=0\).

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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 function's derivatives at a single point, usually around x = 0 (Maclaurin series). It approximates functions using polynomials, where each term involves higher-order derivatives and powers of (x - a).

Binomial Theorem

The binomial theorem provides a formula to expand expressions of the form (1 + u)^n into a sum involving binomial coefficients. For integer powers, it gives exact polynomial expansions, which is useful for finding terms of (1 - x/3)^4 without computing derivatives.

Nonzero Terms in Series

When finding the first four nonzero terms of a series, it is important to identify and include only those terms whose coefficients are not zero. This ensures the approximation captures the significant behavior of the function near the expansion point.

Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.6. (1 - x/3)^4