Key Idea 8.8.1 gives the nth term of the Taylor series of common functions. In Exercises 3 – 6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying pattern.
\[ f(x) = \frac{1}{(1 - x)}; c = 0 \]
==>
\[ f′(x)=\frac{1}{(1 - x)^2}\]
\[ f′′(x)=f′(f′(x))=f′(\frac{1}{(1 - x)^2})=\frac{2}{(1−x)^3}\]
\[ f′′′(x)=f′(f′′(x))=\frac{6}{(1−x)^4}\]
\[ f′′′′(x)=f′(f′′′(x))=\frac{6∗4}{(1−x)^5}\]
The series is bounded by \(x[−1,1]\)
#Loading required package: pracma
require(pracma)## Loading required package: pracma## Warning: package 'pracma' was built under R version 3.6.3#Getting the taylor series
x = function(x) {1/(1-x)}
c = 0
n = 6
p <- taylor(x, c, n)
p## [1] 1.001710 1.000293 1.000029 1.000003 1.000000 1.000000 1.000000