Chapter 8.8
Problem 26:
In Exercise 25-30, use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given function.
First, letโs consider that the Taylor Series of \(f(x)\) centered at c is:
\(\sum _{ n=0 }^{ \infty }{ \frac { { f }^{ n }(c) }{ n! } { (x-c) }^{ n } }\)
\(=f(c)+f'(c)(x-c)+\frac { f''(c) }{ 2! } { (x-c) }^{ 2 }+\frac { f'''(c) }{ 3! } { (x-c) }^{ 3 }+...\)
Solution:
\(f(0)=1\)
\(f'(x)={ -e }^{ -x }\)
\(f'(0)=-1\)
\(f''(x)={ e }^{ -x }\)
\(f''(0)=1\)
\(f'''(x)={ -e }^{ -x }\)
\(f'''(0)=-1\)
\(f''''(x)={ e }^{ -x }\)
\(f''''(0)=1\)
The Taylor series of \({ e }^{ -x }\) with center \(0\): \(1-x+\frac { 1 }{ 2 } { x }^{ 2 }-\frac { 1 }{ 6 } { x }^{ 3 }+\frac { 1 }{ 24 } { x }^{ 4 }+...\)
Thus,
\(\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }\frac { { x }^{ n } }{ n! } }\)