SDoumbia_Week14_TaylorSeriesApproximations

Taylor Series Expansions of Popular Functions

Function \(f(x) = \frac{1}{1-x}\)

The Taylor series expansion of \(f(x) = \frac{1}{1-x}\) around \(x = 0\) is:

fx <- function(x) 1/(1 - x)
taylor_series_1x <- taylor(fx, x0 = 0, n = 7)
print(taylor_series_1x)

## [1] 1.007014 1.001710 1.000293 1.000029 1.000003 1.000000 1.000000 1.000000

Function \(f(x) = e^x\)

The Taylor series expansion for \(f(x) = e^x\) around \(x = 0\).

ex <- function(x) exp(x)
taylor_series_ex <- taylor(ex, x0 = 0, n = 8) 
print(taylor_series_ex)

## [1] 2.505533e-05 1.961045e-04 1.386346e-03 8.334245e-03 4.166657e-02
## [6] 1.666667e-01 5.000000e-01 1.000000e+00 1.000000e+00

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

For \(f(x) = \ln(1 + x)\), the expansion is valid for \(-1 < x \leq 1\).

ln1x <- function(x) log(1 + x)
taylor_series_ln1x <- taylor(ln1x, x0 = 0, n = 6)
print(taylor_series_ln1x)

## [1] -0.1668792  0.2000413 -0.2500044  0.3333339 -0.5000000  1.0000000  0.0000000

Function \(f(x) = x^{1/2}\)

The Taylor series expansion for \(f(x) = x^{1/2}\) around \(x = 1\).

# Manually computing derivatives and Taylor series coefficients because having issues with 'taylor()' function
f <- function(x) sqrt(x)
f1 <- function(x) (1/2) * x^(-1/2)
f2 <- function(x) -(1/4) * x^(-3/2)
f3 <- function(x) (3/8) * x^(-5/2)
f4 <- function(x) -(15/16) * x^(-7/2)
f5 <- function(x) (105/32) * x^(-9/2)
f6 <- function(x) -(945/64) * x^(-11/2)

x0 <- 1 # Easier since sqrt(1) = 1

taylor_series_sqrtx <- c(f(x0), f1(x0), f2(x0)/factorial(2), f3(x0)/factorial(3), f4(x0)/factorial(4), f5(x0)/factorial(5), f6(x0)/factorial(6))

print(taylor_series_sqrtx)

## [1]  1.00000000  0.50000000 -0.12500000  0.06250000 -0.03906250  0.02734375
## [7] -0.02050781