Consider the following function. f(x) = x=², Approximate f by a Taylor polynomial T3(x) with the center x=1. Your answer must be of the form, c₁+c₁ (x−1)+c₂ (x−1)²+c₂(x-1)³ How good is the approximation on the small interval 0.9< x≤ 1.1? Use the Taylor's Inequality to find an upper bound of the error, |R₂(x)|=|f(x)-T3(x)|. (a) (b) (c) Find the Taylor series of f(x) with center at x=1. Use the summation notation whoch displays the n-th term clearly. Then, determine the interval of convergence of this series. Show, by using Taylor Ineqality, that the absolute value of the n-th remainder, R₂(x)\, converges to 0 for every x in the (d) interval [0.6, 1.41. (e) Find a rational expression to which this series converges on its interval of convergence. You must show your systematic procedure instead of simply making a wild guess. Hint: Try differentiation or integration.
Consider the following function. f(x) = x=², Approximate f by a Taylor polynomial T3(x) with the center x=1. Your answer must be of the form, c₁+c₁ (x−1)+c₂ (x−1)²+c₂(x-1)³ How good is the approximation on the small interval 0.9< x≤ 1.1? Use the Taylor's Inequality to find an upper bound of the error, |R₂(x)|=|f(x)-T3(x)|. (a) (b) (c) Find the Taylor series of f(x) with center at x=1. Use the summation notation whoch displays the n-th term clearly. Then, determine the interval of convergence of this series. Show, by using Taylor Ineqality, that the absolute value of the n-th remainder, R₂(x)\, converges to 0 for every x in the (d) interval [0.6, 1.41. (e) Find a rational expression to which this series converges on its interval of convergence. You must show your systematic procedure instead of simply making a wild guess. Hint: Try differentiation or integration.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the following function.
f(x) = x=²,
Approximate f by a Taylor polynomial T3(x) with the center x=1. Your answer must be of the form, cô+c₁ (x−1)+c₂ (x−1)²+c3(x−1)³
How good is the approximation on the small interval 0.9< x≤ 1.1? Use the Taylor's Inequality to find an upper bound of the error,
|R3(x)|=|f(x)-T3(x)|.
(a)
(b)
(c)
Find the Taylor series of f(x) with center at x=1. Use the summation notation whoch displays the n-th term clearly. Then, determine the
interval of convergence of this series.
Show, by using Taylor Inegality, that the absolute value of the n-th remainder, |R₂(x)\, converges to 0 for every x in the
(d)
interval [0.6, 1.41.
(e)
Find a rational expression to which this series converges on its interval of convergence. You must show your systematic
procedure instead of simply making a wild guess. Hint: Try differentiation or integration.
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