Q8. (a) Define the radius of convergence of a complex power series an(b) Compute the radius of convergence of the power series31/12Σ(-1)", -z²n+1.2n + 3n=1∞∞n=0

Here we have to give defination of complex power series and radius of convergence of given power…

Answered: Q8. (a) Define the radius of…

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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P1 (a) Use the integral test to study the convergence of the series In(n) n² n=2 Edit: It previously said log(r) instead of ln(x). (b) Use the alternating test to study the convergence of the series 8 Σ(-1)" n=1 (c) Study the convergence of the following series n=1 2n n! (−1)n+1 2n + (−1)n Is it absolutely convergent? Remember to justify all the hypothesis needed to use the test in both parts of this exercise.
Show all of your work and mention what formula/theorem you are using. (a) We are going to find a formula for the sum of the series i(i +1) i=1 1 1 i(i + 1) i+1" First observe that - (i) Find a closed form formula for the nth partial sum s, in terms of n (but without any sigma). (ii) Find lim sp. (b) Consider the series (i) Use properties of logarithm to find a closed form formula for the nth partial sum in terms of n (but without any sigma). (ii) Does the series converge or diverge?
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
(-1)"-1 6. (a) Show that the series is convergent. n2 n=1 (-1)"-1 (b) Find the partial sum s, of the series Estimate the error in using s; as an n2 n=1 approximation to the sum of the series. (c) Find a value of n so that s, is within 0.001 of the sum.
(11)

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Q8. (a) Define the radius of convergence of a complex power series an. (b) Compute the radius of convergence of the power series 3/7/20 2n + 3 ∞ Σ(-1)", n=1 ∞ 2n+1 2 n=0