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{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
sin x ≈ x − x³/6 on [π/4, π/4]
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.2.8
Suppose the power series ∑ₖ₌₀∞ cₖ(x−a)ᵏ has an interval of convergence of (−3,7]. Find the center a and the radius of convergence R.
Guida verificata passo dopo passo1
Recall that the interval of convergence of a power series \( \sum_{k=0}^\infty c_k (x - a)^k \) is centered at \( a \), and the radius of convergence \( R \) is the distance from \( a \) to either endpoint of the interval.
Given the interval of convergence \( (-3, 7] \), identify the center \( a \) as the midpoint of the interval endpoints. Calculate \( a = \frac{-3 + 7}{2} \).
Calculate the radius of convergence \( R \) as the distance from the center \( a \) to either endpoint. Use \( R = 7 - a \) or \( R = a - (-3) \).
Verify that the radius \( R \) is the same when calculated from both endpoints to confirm consistency.
Summarize the results: the center \( a \) and the radius of convergence \( R \) define the power series' interval of convergence.
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Power Series and Center
A power series is an infinite sum of terms in the form cₖ(x−a)ᵏ, where a is the center of the series. The center a is the point around which the series is expanded, and it determines the horizontal shift of the interval of convergence.
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05:58
Intro to Power Series
Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges. It is typically an interval centered at a, possibly including or excluding endpoints, and is crucial for understanding where the series represents a valid function.
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08:44Interval of Convergence
Radius of Convergence
The radius of convergence R is the distance from the center a to either endpoint of the interval of convergence. It measures how far from a the series converges and is calculated as half the length of the interval of convergence.
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07:36Radius of Convergence
Pratica correlata
Domanda del libro di testo
Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
{(eˣ−1)/x if x ≠ 1, 1 if x = 1
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How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?
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