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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
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모든 교과서Hass 15th EditionCh. 10 - Infinite Sequences and Series문제 10.9.33
10장, 문제 10.9.33

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²

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Recall that the Maclaurin series is the Taylor series expansion of a function about \(x=0\). We want to find the first four nonzero terms of the Maclaurin series for the function \(f(x) = (\tan^{-1} x)^2\).
Start by writing the Maclaurin series for \(\tan^{-1} x\). The known expansion is: \[ \tan^{-1} x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \]
Since \(f(x) = (\tan^{-1} x)^2\), square the series for \(\tan^{-1} x\) up to the terms that will produce the first four nonzero terms in \(f(x)\). This means considering terms up to at least \(x^7\) in \(\tan^{-1} x\) to get terms up to \(x^8\) in \(f(x)\).
Perform the multiplication: \[ \left(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \right)^2 \] Multiply out the terms carefully, combining like powers of \(x\), and keep track of the coefficients for each power.
Collect the terms by powers of \(x\) and identify the first four nonzero terms in the resulting series. These will be the first four nonzero terms of the Maclaurin series for \((\tan^{-1} x)^2\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about zero. It expresses the function as an infinite sum of terms involving derivatives evaluated at zero, allowing approximation near x = 0. Understanding how to find and use these series is essential for expanding functions like (tan⁻¹x)².
추천 영상:
08:26
Convergence of Taylor & Maclaurin Series

Series Expansion of Inverse Trigonometric Functions

The inverse tangent function, tan⁻¹x, has a known Maclaurin series expansion involving odd powers of x with alternating signs. Recognizing and using this series is crucial for finding the expansion of (tan⁻¹x)² by substituting and manipulating the series terms.
추천 영상:
06:35
Derivatives of Other Inverse Trigonometric Functions

Multiplication and Collection of Series Terms

To find the series for (tan⁻¹x)², one must square the Maclaurin series of tan⁻¹x and combine like powers of x. This involves multiplying series term-by-term and carefully collecting coefficients of each power to identify the first four nonzero terms.
추천 영상:
가이드 코스
06:00
Geometric Series
관련 실천
교과서 질문

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 ∞) (1 + cos n) / n²

교과서 질문

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 ∞) 1 / (1 + 2² + 3² + ⋯ + n²)

교과서 질문

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=3 to ∞) [2n² / n²ⁿ]

교과서 질문

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 1 to ∞) tan(n^(1/n))

교과서 질문

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [n! / (2n + 1)!]

교과서 질문

In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.

f(x) = (sin x) ln(1 + x)