IS 605 - Assignment 14

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.

\[f(x) = \frac{1}{1-x}\] \[f(x) = e^x\] \[f(x) = ln(1 + x)\]

Any function that is infinitely differentiable can be represented by a polynomial of the following form:

\[f(x) = \sum_{n=0}^\infty \frac{f^{n}(a)}{n!}(x-a)^{n}\]

where \(f^{n}(a)\) represents the \(nth\) derivative of \(f(x)\) evaluated at \(x = a\).

Problem 1

\[f(x) = \frac{1}{1-x}, x\in(-1, 1)\] \[= \sum_{n=0}^\infty x^{n}, x\in(-1, 1)\]

\[= 1 + x + x^{2} + x^{3} + x^{4} + x^{5} +...\]

library(ggplot2)

p <- ggplot(data = data.frame(x=0), mapping = aes(x = x))
fun.1 <- function(x) 1/(1-x)
taylor.1 <- function(x) 1 + x + x^2 + x^3 + x^4 + x^5

p + 
  layer(stat = "function", 
    fun = fun.1, 
    mapping = aes(color = "fun.1"),
    position = "jitter"
    ) + 
  layer(stat = "function", 
    fun = taylor.1, 
    mapping = aes(color = "taylor.1")
    ) + 
  scale_x_continuous(limits = c(-1, 1)) +
  scale_color_manual(name = "Functions", 
                     values = c("blue", "red"),
                     labels = c("1/(1-x)", "Taylor Series Approximation (5th order)"))

Problem 2

\[f(x) = e^x\] \[= \sum_{n=0}^\infty \frac{x^{n}}{n!}, x\in(-1, 1)\]

\[= 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120} +...\]

p <- ggplot(data = data.frame(x=0), mapping = aes(x = x))
fun.2 <- function(x) exp(x)
taylor.2 <- function(x) x^0/factorial(0) + x^1/factorial(1) + x^2/factorial(2) + x^3/factorial(3) + x^4/factorial(4) + x^5/factorial(5)

p + 
  layer(stat = "function", 
    fun = fun.2, 
    mapping = aes(color = "fun.2"),
    position = "jitter"
    ) + 
  layer(stat = "function", 
    fun = taylor.2, 
    mapping = aes(color = "taylor.2")
    ) + 
  scale_x_continuous(limits = c(-1, 1)) +
  scale_color_manual(name = "Functions", 
            values = c("blue", "red"),
            labels = c("e^x", "Taylor Series Approximation (5th order)"))

Problem 3

\[f(x) = ln(1 + x)\] \[= \sum_{n=0}^\infty (-1)^{n+1}\frac{x^{n}}{n}, x\in(-1, 1)\]

\[= x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} -...\]

p <- ggplot(data = data.frame(x=0), mapping = aes(x = x))
fun.3 <- function(x) log(1+x)
taylor.3 <- function(x) x - (x^2/2) + (x^3/3) - (x^4/4) + (x^5/5)

p + 
  layer(stat = "function", 
    fun = fun.3, 
    mapping = aes(color = "fun.3"),
    position = "jitter"
    ) + 
  layer(stat = "function", 
    fun = taylor.3, 
    mapping = aes(color = "taylor.3")
    ) + 
  scale_x_continuous(limits = c(-1, 1)) +
  scale_color_manual(name = "Functions", 
            values = c("blue", "red"),
            labels = c("log(1+x)", "Taylor Series Approximation (5th order)"))