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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.10.1
Chapter 10, Problem 10.10.1

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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1
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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Key Concepts

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

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).
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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.
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Fundamental Theorem of Calculus Part 1

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.
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Related Practice
Textbook Question

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

2, 6, 10, 14, 18, …Every other even positive integer

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Textbook Question

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

Textbook Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]

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Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

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Textbook Question

Series with Geometric Terms

In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.

∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]

Textbook Question

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 2 + (-1)ⁿ

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