2. Let s (0, ∞) → R be the usual square root defined on the nonnegative reals: s(x) is the uniquenonnegative real number t such that t² = x.(a) Give an expression for the derivatives(k) (1) =in the complex variable z.dkdrkk=0s(x)for an arbitrary ke N. You may use knowledge from previous courses in real calculus.(b) Compute the radius of convergence of the seriesx=1s(k) (1) z kk!

Answered: Image /qna-images/answer/7928ebf2-4c10-46a8-be13-0674a09df0d9.jpg

Answered: . Let s (0, ∞) → R be the usual square…

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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Use interval notation to present the answers to these three questions. (a) Find the set of real numbers t for which the following series converges: A(t) = (9-18t)". n=7 Answer: (b) Find the set of all real numbers a for which the following series converges: ∞ B(a)=)|arctan (13)). 73 n=4 Answer: ((-73tan(1))/14, (73tan(1))/14) (c) Find the largest interval including 0 on which the following series converges: C(0)=2" sin 2n 7-4 Answer: 21 124 0
q4
Suppose ∞ 3n f(x) = Σ n=0 To determine X n! 4 [ se f(x) dx 3.9 Question Help: (x - 4)" 4 3.9 (a) da to within 0.0001, it will be necessary to add the first syntax incomplete. terms of the series. (Enter the answer accurate to four decimal places) Video Written Example Message instructor
le) (b) 2E" 2" 12. Check the convergence of the following power series, find the radius and interval of conver- gence: ( a) Σ" (x+ n)" ( b) Σ Also find the sum as a function of x for this problem. 13. Find the Taylor series generated by ƒ at r = a and the Maclaurin series: (a) f(x) = 1/x², a = 1 (b) f(x) = xª + x² + 1, a = -2 14. Find the Taylor polynomials of order 1,2,3 for the function f at a: (a) f(x)= /T, a = 4 (b) f(r) = cos r, a = r /4 15. (a) Calculate e with an error of less than 10-º. (b) Estimate the error in the approximation sinh(x) = x + (r"/3!) when |r| < 0.5. (c) How close is the approximation sin(x) = r when |æ| < 10¬³? For which of these values of r is a< sin(x)? %3D
6.5 Use the series for f(x) = on [x| < 1 to construct a series for (1 – x)(x – 2)' Determine the interval of convergence.
1. For the given the power series find the radius of convergence R and the interval of convegence I. (-1)(x-1)" (a) Σ1 n(n+1) (b) ΣΩ (0) Σαο (1)"x" n! (-1) 2n 3m
Q1 Evaluate the limit by expressing the functions involved as a Taylor Series. 1-cost-(t²/2) t4 (Remember there are some algebra involved!) lim t->0
(11)

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