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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 10.8.2
Chapter 10, Problem 10.8.2

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) = sin x,a = 0

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1
Identify the function and the point of expansion: here, the function is \(f(x) = \sin x\) and the Taylor polynomials are centered at \(a = 0\).
Recall the general formula for the Taylor polynomial of order \(n\) centered at \(a\): \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k,\] where \(f^{(k)}(a)\) is the \(k\)-th derivative of \(f\) evaluated at \(a\).
Calculate the derivatives of \(f(x) = \sin x\) up to the third order and evaluate each at \(x = 0\): - \(f(x) = \sin x\), - \(f'(x) = \cos x\), - \(f''(x) = -\sin x\), - \(f'''(x) = -\cos x\). Then find \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f'''(0)\).
Write out each Taylor polynomial using the derivatives evaluated at 0: - Order 0: \(T_0(x) = f(0)\), - Order 1: \(T_1(x) = f(0) + f'(0)x\), - Order 2: \(T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\), - Order 3: \(T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3\).
Simplify each polynomial by substituting the values of the derivatives at 0 and factorials, leaving the expressions in terms of powers of \(x\).

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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 its derivatives at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n grows.
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Taylor Polynomials

Derivatives of Sine Function

The sine function has a cyclic pattern in its derivatives: f(x) = sin x, f'(x) = cos x, f''(x) = -sin x, f'''(x) = -cos x, and so on. Knowing these derivatives at a specific point is essential for constructing the Taylor polynomial.
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Derivatives of Sine & Cosine

Evaluating Derivatives at the Expansion Point

To build the Taylor polynomial at a point a, each derivative of the function must be evaluated at a. For a = 0, this means calculating sin(0), cos(0), and their higher derivatives at zero, which simplifies the polynomial coefficients.
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Related Practice
Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (ln n) xⁿ ]

Textbook Question

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

1
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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) (x + 5)ⁿ

Textbook Question

Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

1
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Textbook Question

Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

29. lim (x → 0) (e^x - (1 + x)) / x²