wk14_assignment

library(pracma)

ASSIGNMENT 14 - TAYLOR SERIES

This week, we’ll work out some Taylor Series expansions of popular functions

f(x)=1/(1−x)

f′(x)=1/(1−x)2

f″(x)=f′(f′(x))=f′(1/(1−x)2)=2/(1−x)3x)3

f‴(x)=f′(f″(x))=6/(1−x)4

f⁗(x)=f′(f‴(x))=6∗4/(1−x)5

The series is bounded by x[−1,1]

1. f(x)=1/(1−x)

tylr_ser <- function(x) {1/(1-x)}
# Find the Taylor series of q1(x) centered at x=0 up to 6th degree
(taylor(tylr_ser, 0, 6))

## [1] 1.001710 1.000293 1.000029 1.000003 1.000000 1.000000 1.000000

2. f(x) = \(e^x\)

tylr_expo <- function(x) {exp(x)}
(taylor(tylr_expo, 0, 4))

## [1] 0.04166657 0.16666673 0.50000000 1.00000000 1.00000000

3. f(x) = ln(1+x)

tylr_ln <- function(x) {log(1+x)}
(taylor(tylr_ln, 0, 4))

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

4. f(x) = \(x^{\frac {1}{2}}\)

tylr_frac <- function(x){x^1/2}
(taylor(tylr_frac, 0, 4))

## [1] 0.5 0.0