There are two parts to this problem and I need help with both: Using the Taylor series for ex A) I need to write out the first four terms of the Taylor series about 0 for f(x) = e-3x using the existing series. B) I need to differentiate the Taylor series found in Part A for the function  f(x) = e-3x and rewrite that series in summation notation.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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There are two parts to this problem and I need help with both:

Using the Taylor series for ex

A) I need to write out the first four terms of the Taylor series about 0 for

f(x) = e-3x using the existing series.

B) I need to differentiate the Taylor series found in Part A for the function 

f(x) = e-3x and rewrite that series in summation notation.

Expert Solution
Step 1

f(x)=ex

To find first four terms of Taylor series at a=0

f(0)+f'(0)(x-0)1!+f''(0)(x-0)22!+f'''(0)(x-0)33!

f(x)=exf'(x)=exf''(x)=exf'''(x)=exf(0)=e(0)=1f'(0)=1f''(0)=1f'''(0)=1

Step 2

f(0)+f'(0)(x-0)1!+f''(0)(x-0)22!+f'''(0)(x-0)33!

Replace all the values 

1+1x1!+1x22!+1x33!ex=1+x+x22+1x32

Put x=-3x

e-3x=1+-3x+-3x22+-3x32e-3x=1-3x+9x22-27x32

 

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