Data 605 - Homework #14

library(pracma)

$f(a)+f^{\mathrm{\prime <!--\; \text{'}\; -->}}(a)(x-<!--\; ???\; -->a)+\frac{f^{\mathrm{\prime <!--\; \text{'}\; -->}\mathrm{\prime <!--\; \text{'}\; -->}}}{2!}(x-<!--\; ???\; -->a)+\frac{f^{\mathrm{\prime <!--\; \text{'}\; -->}\mathrm{\prime <!--\; \text{'}\; -->}\mathrm{\prime <!--\; \text{'}\; -->}}}{3!}(x-<!--\; ???\; -->a)+\frac{f^{(4)}}{4!}(x-<!--\; ???\; -->a)+...$

$1+x+x^2+x^3+x^4+x^5+O\left(x^6\right)$

Taylor_equation1 <- function(x) {1/(1-x)}
p <- taylor(Taylor_equation1, x0 = 0, n = 4)
p

## [1] 1.000029 1.000003 1.000000 1.000000 1.000000

$1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+O\left(x^6\right)$

Taylor_equation2 <- function(x) {exp(x)}
p <- taylor(Taylor_equation2, x0 = 0, n = 4)
p

## [1] 0.04166657 0.16666673 0.50000000 1.00000000 1.00000000

$x-<!--\; ???\; -->\frac{x^2}{2}+\frac{x^3}{3}-<!--\; ???\; -->\frac{x^4}{4}+\frac{x^5}{5}-<!--\; ???\; -->\frac{x^6}{6}+O\left(x^7\right)$

Taylor_equation3 <- function(x) {log(1+x)}
p <- taylor(Taylor_equation3, x0 = 0, n = 4)
p

## [1] -0.2500044  0.3333339 -0.5000000  1.0000000  0.0000000