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For the function f(x) = In (1 + 2x), find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a = 0.

For the function f(x) = In (1 + 2x), find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a = 0.

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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For the function f(x) = In (1 + 2x), find the Taylor polynomials of orders 0, 1, 2, and 3 generated by fat a = 0. Po(x) =
Transcribed Image Text:For the function f(x) = In (1 + 2x), find the Taylor polynomials of orders 0, 1, 2, and 3 generated by fat a = 0. Po(x) =
Expert Solution
Step 1: Finding the Taylor series expansion

The Taylor series expansion of fa+x is given by 

    fa+x=fa+xf'a+x22f"(a)+x33!f'''(a)+...+xn-1n-1!fn-1a+Rna=0⇒      fx=fa+xf'0+x22f"(0)+x33!f'''(0)+...+xn-1n-1!fn-10+Rn     ....i

Given fx=ln1+2x

Now 

   fx=ln1+2x⇒f0=0  f'x=21+2x⇒f'0=2    f"(x)=-41+2x2⇒  f"(0)=-4     f'''x=161+2x3⇒f'''x=16

fx=ln1+2x

Taylor series polynomial of order n=0

Using formula 1

fx=0

Taylor series polynomial of order n=1

Using formula 1

     fx=0+2x⇒fx=2x

Taylor series polynomial of order n=2

Using formula 1

fx=0+2x+x22-4       =2x-2x2

Taylor series polynomial of order n=3

Using formula 1

fx=0+2x-2x2+x3616      =2x-2x2+83x3

 

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