Discusison 14

26: use the Taylor series given in Key Idea

8.8.1 to create the Taylor series of the given functons.

26. \(f(x) = e^{-x}\)

f <- function(x) {
  return(exp(-x))
}

taylor_series <- function(x, n) {
  sum <- 0
  for (i in 0:n) {
    sum <- sum + (f(i) / factorial(i)) * x^i
  }
  return(sum)
}

x_values <- c(0, 1, 2, 3)

n_values <- c(1, 2, 3, 4)

for (x in x_values) {
  cat("x =", x, "\n")
  for (n in n_values) {
    cat("n =", n, ": ", taylor_series(x, n), "\n")
  }
  cat("\n")
}

## x = 0 
## n = 1 :  1 
## n = 2 :  1 
## n = 3 :  1 
## n = 4 :  1 
## 
## x = 1 
## n = 1 :  1.367879 
## n = 2 :  1.435547 
## n = 3 :  1.443845 
## n = 4 :  1.444608 
## 
## x = 2 
## n = 1 :  1.735759 
## n = 2 :  2.006429 
## n = 3 :  2.072812 
## n = 4 :  2.085023 
## 
## x = 3 
## n = 1 :  2.103638 
## n = 2 :  2.712647 
## n = 3 :  2.936689 
## n = 4 :  2.998504