This week, we’ll work out some Taylor Series expansions of popular functions.For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion.

\(f\left( x \right) \quad =\quad \sum _{ n=0 }^{ \infty }{ \frac { { f }^{ (n) }(c) }{ n! } { (x-c) }^{ n }\quad on\quad I.\quad \quad \quad \quad (f(x)\quad converges\quad if\quad } { lim }_{ n->\infty }{ R }_{ n }(x)\quad =\quad 0)\)

#install.packages("pracma")  
library(pracma)
## Warning: package 'pracma' was built under R version 3.4.3
#install.packages("jpeg")  
#library(jpeg)
  1. $f( x ) = $ Anser:
# read answer1 image
#img1 <- readJPEG("answer1.jpg")
#img1 <- readJPEG(system.file("img", "answer1.jpg", package="jpeg"))

f1<- function(x) { 1/(1-x)}
taylor_f1 <- taylor(f1,0,5)

x1<- seq(-1.0, 1.0, length.out = 50)
y1 <- f1(x1)

yr_taylor <- polyval(taylor_f1, x1)
plot(x1, y1, type = "l", main= '1/(1-x) Taylor Series' ,col = "lightblue", lwd = 3)

  1. \(f\left( x \right) \quad =\quad { e }^{ x }\) Answer:
f2<- function(x) { 
  e<-exp(1) 
  return(e^x) }
taylor_f2 <- taylor(f2,0,5)

x2<- seq(-1.0, 1.0, length.out = 1500)
y2 <- f2(x2)

yr_taylor <- polyval(taylor_f2, x2)
plot(x2, y2, type = "l", main= 'e^x Taylor Series' ,col = "lightblue", lwd = 3)

  1. \(f\left( x \right) \quad =\quad ln(1+x)\) Answer:
f3<- function(x) {log(1+x)}
taylor_f3 <- taylor(f3,0,5)

x3<- seq(-1.0, 1.0, length.out = 1500)
y3 <- f3(x3)

yr_taylor <- polyval(taylor_f3, x3)
plot(x3, y3, type = "l", main= 'ln(1+x) Taylor Series' ,col = "lightblue", lwd = 3)