Skip to main content
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 10.8.18
Chapter 10, Problem 10.8.18

Finding Taylor Series at x = 0 (Maclaurin Series)
Find the Maclaurin series for the functions in Exercises 11–24.
5 cos πx

Verified step by step guidance
1
Recall that the Maclaurin series is the Taylor series expansion of a function at \(x = 0\), given by the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\] where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Identify the base function inside the given function. Here, the function is \(f(x) = 5 \cos(\pi x)\). We can factor out the constant 5 and focus on the Maclaurin series for \(\cos(\pi x)\).
Recall the Maclaurin series for \(\cos u\) is \[\cos u = \sum_{n=0}^{\infty} (-1)^n \frac{u^{2n}}{(2n)!}.\] In this problem, substitute \(u = \pi x\) to get \[\cos(\pi x) = \sum_{n=0}^{\infty} (-1)^n \frac{(\pi x)^{2n}}{(2n)!}.\]
Multiply the entire series by 5 to account for the coefficient in the original function: \[5 \cos(\pi x) = 5 \sum_{n=0}^{\infty} (-1)^n \frac{(\pi x)^{2n}}{(2n)!} = \sum_{n=0}^{\infty} 5 (-1)^n \frac{\pi^{2n} x^{2n}}{(2n)!}.\]
Write out the first few terms explicitly by plugging in \(n=0,1,2,\ldots\) to see the pattern of the series, which helps in understanding the expansion and verifying correctness.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Maclaurin Series

A 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 and divided by factorial terms. This series helps approximate functions near zero.
Recommended video:
08:26
Convergence of Taylor & Maclaurin Series

Derivatives of Trigonometric Functions

To find the Maclaurin series of 5 cos(πx), you need to compute derivatives of cos(πx) at x = 0. The derivatives cycle every four steps due to the periodic nature of cosine and sine, and each derivative involves the chain rule because of the inner function πx.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions

Chain Rule in Differentiation

The chain rule is used to differentiate composite functions like cos(πx). It states that the derivative of f(g(x)) is f'(g(x)) times g'(x). Applying this rule correctly is essential to find the derivatives needed for the Maclaurin series coefficients.
Recommended video:
05:02
Intro to the Chain Rule
Related Practice
Textbook Question

Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.

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

1
views
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)]

1
views
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)

1
views
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

1
views