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(a) Evaluate the respective sum and product. 4 2" (i) E en É 2n+2 (i) П n=0 п n=1

(a) Evaluate the respective sum and product. 4 2" (i) E en É 2n+2 (i) П n=0 п n=1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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As a refresher evaluate analytically the following expressions. Report your answer in simplest form. (a) Evaluate the respective sum and product. 4 5 (1) È„" en (ii) II 2n +2 n=0 n=1 (b) Evaluate the following derivatives (Note. sinc(x) = sin(x)/x). d2 (ii) d (#) Inle"| d Inle*| (i) sinc(x) dx dx (c) Evaluate the following integrals. 27 * dx хе (ii) * cos(x)dx +i sin(x)dx (i) (d) Recall if a function f has a power series expansion at x = a, then its Taylor Series represen- tation is f(1) (a) f(2) (a) f(3) (a) 00 f (x) = f (a) (x – a)" = f(O) (a) + (x – a) + (@) (x – a)² + (x– a)³ + · · · (1) 3! n! 1! 2! n=0 d" f where f(n) series, dxn • If f can be expanded at x = 0 and a = 0, the series is called a Maclaurin f (x) = (0)g" = f(0)(0) + f(1) (0) f(2) (0) ,2 + f(3) (0)3 -x + 1! +. (2) .. n! 2! 3! n=0 Using Eq. 2 derive the Maclaurin representation of x2n+1 sin(x) = E(-1)". (2n + 1)! n=0
Transcribed Image Text:As a refresher evaluate analytically the following expressions. Report your answer in simplest form. (a) Evaluate the respective sum and product. 4 5 (1) È„" en (ii) II 2n +2 n=0 n=1 (b) Evaluate the following derivatives (Note. sinc(x) = sin(x)/x). d2 (ii) d (#) Inle"| d Inle*| (i) sinc(x) dx dx (c) Evaluate the following integrals. 27 * dx хе (ii) * cos(x)dx +i sin(x)dx (i) (d) Recall if a function f has a power series expansion at x = a, then its Taylor Series represen- tation is f(1) (a) f(2) (a) f(3) (a) 00 f (x) = f (a) (x – a)" = f(O) (a) + (x – a) + (@) (x – a)² + (x– a)³ + · · · (1) 3! n! 1! 2! n=0 d" f where f(n) series, dxn • If f can be expanded at x = 0 and a = 0, the series is called a Maclaurin f (x) = (0)g" = f(0)(0) + f(1) (0) f(2) (0) ,2 + f(3) (0)3 -x + 1! +. (2) .. n! 2! 3! n=0 Using Eq. 2 derive the Maclaurin representation of x2n+1 sin(x) = E(-1)". (2n + 1)! n=0
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