TAYLOR SERIES

For each function, only consider its valid ranges as indicated in the notes when you arecomputing the Taylor Series expansion.

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

Equation 1

\(f\left( x \right) = \frac { 1 }{ (1-x) }\)
\(f(a)\quad +{ \quad f }^{ (1) }(a)(x-a)\quad +\quad \frac { { f }^{ (2) } }{ 2! } (a)(x-a)\quad +\quad ...\)

Derivatives for \(f\left( x \right) = \frac { 1 }{ (1-x) }\) are:

\(f'(x) = \frac{1}{(1-x)^2}\)
\(f''(x) = \frac{2}{(1-x)^3}\)
\(f'''(x) = \frac{6}{(1-x)^4}\)

\(f(a) + {{ f }^{ \prime }}(a)(x-a) + \frac{{ f }^{ \prime \prime }}{2!}(x-a) + \frac {{ f }^{ \prime \prime \prime }}{3!}(x - a) + \frac {f^{(4)}}{4!}(x - a) +...\)

\(= 1 + 1x + \frac{2}{2!}x^2 + \frac{6}{3!}x^3 + \frac{24}{4!}x^4 +...\)

\(1 + x + x^2 + x^3 + x^4 + …….\)

library(pracma)
eq <- function(x) {1/(1-x)}
p <- taylor(eq, x0 = 0, n = 4)
p
## [1] 1.000029 1.000003 1.000000 1.000000 1.000000

Equation 2

\(f(x) = e^x\)
\(f(a) \quad= \quad { e }^{ a }\)
\(f(a)' \quad= \quad { e }^{ a }\)
\(f(a)'' \quad= \quad { e }^{ a }\)
\(f(a)''' \quad= \quad { e }^{ a }\)
\(f(a)'''' \quad= \quad { e }^{ a }\)
equation <- function(x) {exp(x)}
q <- taylor(equation, x0 = 0, n = 4)
q
## [1] 0.04166657 0.16666673 0.50000000 1.00000000 1.00000000

Equation 3

\(f(x) = ln(1 + x)\)

\(f(a) \quad= \quad ln(1+a)\) \(\quad= \quad f(0) = 0\)

\({ f }^{ \prime }(a) \quad= \quad \frac{1}{1+a}\) \(\quad= \quad f(0) = 1\)

\({ f }^{ \prime \prime }(a) \quad= \quad \frac{-1}{(1+a)^2}\) \(\quad= \quad f(0) = -1\)

\({ f }^{ \prime \prime \prime }(a) \quad= \quad \frac{2}{(1+a)^3}\) \(\quad= \quad f(0) = 2\)

\(f^{(4)}(a) \quad= \quad \frac{-6}{(1+a)^4}\) \(\quad= \quad f(0) =-6\)
eq1 <- function(x) {log(1+x)}
r <- taylor(eq1, x0 = 0, n = 4)
r
## [1] -0.2500044  0.3333339 -0.5000000  1.0000000  0.0000000