Textbook Question
Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.8.10
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
Verified step by step guidance1
Identify the function and the point about which the Taylor polynomial is generated: here, \( f(x) = \sqrt{1 - x} \) and \( a = 0 \).
Calculate the derivatives of \( f(x) \) up to the third order: find \( f'(x) \), \( f''(x) \), and \( f'''(x) \).
Evaluate the function and its derivatives at \( x = 0 \): compute \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f'''(0) \).
Write the Taylor polynomials of orders 0, 1, 2, and 3 using the formula: \[ T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k \], substituting \( a = 0 \) and the evaluated derivatives.
Express each polynomial explicitly: the 0th order is just \( f(0) \), the 1st order adds the linear term, the 2nd order adds the quadratic term, and the 3rd order adds the cubic term, all centered at \( x = 0 \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Polynomials
Taylor polynomials approximate a function near a point a by using a finite sum of derivatives of the function at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n increases.
Recommended video:
07:00
Taylor Polynomials
Derivatives of Functions
Derivatives measure the rate of change of a function. To find Taylor polynomials, you must compute the function's derivatives at the point a, as these derivatives determine the coefficients of the polynomial terms.
Recommended video:
06:30Derivatives of Other Trig Functions
Evaluating at the Expansion Point
The Taylor polynomial is centered at a specific point a, where the function and its derivatives are evaluated. This evaluation ensures the polynomial matches the function's value and slope behavior at that point, making the approximation accurate near a.
Recommended video:
5:14Evaluate Logarithms
Related Practice
Textbook Question
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
1
views
Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ
1
views
Textbook Question
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
1
views
Textbook Question
In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.
1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …
1
views
Textbook Question
Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]