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12. Check the convergence of the following power series, find the radius and interval of conver- gence: (a) Σn"z" (x + 1)" (b) Σ √₁ (0) Ỹ ((² + ¹))" Also find the sum as a function of x for this problem.

12. Check the convergence of the following power series, find the radius and interval of conver- gence: (a) Σn"z" (x + 1)" (b) Σ √₁ (0) Ỹ ((² + ¹))" Also find the sum as a function of x for this problem.

Advanced Engineering Mathematics
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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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
Transcribed Image Text: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
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