Please see attached image... Let R,(x) be the remainder of the Taylor expansion of e":er" +1R, (x)for some c e (0, x)(n + 1)!'Show that for any fixed x, lim Rn(x) = 0n+ 00

Introduction: Suppose a function fx is continuously differentiable at a point over a certain…

Answered: Let R(x) be the remainder of the Taylor…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.1: Sequences And Their Notations

Problem 71SE: Prove the conjecture made in the preceding exercise.

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Prove the conjecture made in the previous exercise.
Below is a graph of function f(x). Which of the following could be the second Taylor polynomial of f(x) at x=a? fx) O p20) =-a) 13 %3D P20) =D을 (x-a)-를 (x- a)2 13 p2(x) = P2(x) = - a) +*-a)? (x-) P2() = (x - a)2
Suppose f and g are C" functions with Taylor expansions denoted Tn(f, xo, x) and Ta(9, xo, x). Prove that Tn(f,20, r)+Tn(9, xo, x) is the Taylor expansion of f + g at xo.
Suppose a functions satisfies f(5) = 2, f'(5) = 5, f"(5) = -7, and %3D f''(5) = 12. What is the third degree Taylor polynomial for f about x = 5? A) 2+ 5(x - 5) -7(x - 5)2 +12(x - 5)3 7 B) 2+5(x – 5) –(x- 5)² + 2(x – 5)³ 2 a 245x-+2x x²+2x³ D) 10+25x - 35 x²+6x3
39.. Find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. 1 f (x)= 1+x? a. S2(x)=1-x² s,(x)= 1-x²+x* S (x) = 1-x²+x+-x S(x)= 1-x²+xt-x°+x³ b. S2(x) = 1+x² S,(x)=1+x²+x* S(x)= 1+x²+x++x Sx)= 1+x²+x*+x°+x° c. S,(x)= 1-x² S,(x)=1-x²-x S,(x) = 1-x²-x+-x° S;(x)= 1-x²-x+-x°-x° S(x) = 1+x²-x++x d. S,(x)= 1+x² S,(x) = 1+x²-x* e. S,(x)= 1-x² S,(x)=1-x²+x* S(x)= 1-x²+x++x S,(x)=1-x²+x++x°+x® * $x)= 1+x²-x*+x°-x³
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Find the nth Taylor polynomial for the function, centered at c. f(x) = In(x), n = 4, C = 7 %3D (n – 7)* ( – Pa0 - In(7) + (n - 7)( }) + (*(- 1) + (° () + (a-2)"(- ) – 7)* ( 3 7)()+ (п—7)2 PĄ(x) = In(7)+ (n – 7) 49 3! 343 4! 2401
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Which of the following is the approximate value of e' if polynomial expansion P(x) centered at a = 0 of e* is used? A.1 +e + + 2. B. 1+e + +-+ 3. 4 + 24 120 720 C. 2+-+- ++ 24 120 1 D. 2 +- + - +-+-+ 2 720 3 4 + -10
a) Which of the following is the correct definition for the nth Taylor Polynomial for the function f (x) at x = a? [ Select ] f" (a) Option I. Pn (x) = f (a) + (x – a) + ( – a)? + a)°+. (n+1)! 2 - a)" Option II. Pn (2) = f (x) + f' (x) (x - a) + ( – a)² + ( – a)³ +.. (- a)" Option III. P. (2) = f(a) + f' (a) (x - a) + ( – a)² + (x – a)³ +....+ (x- a)" Option VI. Pn (a) = f (a) + f' (a) (x – a) + (2 – a') + (2 – a') +....+ (x – a") b) Suppose now that f (x) is some differentiable function such that f (2) = 10 and that the nth derivative of f (x) at x = 2 is f(m) (2) = 2" for 1

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Let R(x) be the remainder of the Taylor expansion of e: e®z" +1 (n + 1)!' R,(x) for some c e (0, z) Show that for any fixed z, lim Rn(x) = 0