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(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(M) (a) (x – a)" = f(0 (a) + f(2) (a), f(3) (a), 00 f(x) = E n! a) + (x- a)² + (x – a)³ + · · · (1) 3! 1! 2! + (v – x). n=0 where f(n) = . If f can be expanded at x = 0 and a = series, 0, the series is called a Maclaurin dx • f(w) (0), f(1) (0) f(2) (0). f(3) (0) 00 f(x) = E x" = f(0) (0) + n! -x + 1! (2) 2! 3! ...+ *- n=0 Using Eq. 2 derive the Maclaurin representation of x2n+1 Sin (x )-Σ(-1)", 00 (2n + 1)! n=0

(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(M) (a) (x – a)" = f(0 (a) + f(2) (a), f(3) (a), 00 f(x) = E n! a) + (x- a)² + (x – a)³ + · · · (1) 3! 1! 2! + (v – x). n=0 where f(n) = . If f can be expanded at x = 0 and a = series, 0, the series is called a Maclaurin dx • f(w) (0), f(1) (0) f(2) (0). f(3) (0) 00 f(x) = E x" = f(0) (0) + n! -x + 1! (2) 2! 3! ...+ *- n=0 Using Eq. 2 derive the Maclaurin representation of x2n+1 Sin (x )-Σ(-1)", 00 (2n + 1)! n=0

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.1: Tables And Trends
Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...
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I need help with part d. I want to see detailed step.

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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