Question 1

Compute the taylor series (to order 4) of: \[\begin{aligned} f(x) = \frac{1}{1-x} \end{aligned}\]

We can use the taylor function included in the calculus library within R for this. The above functon is valid everywhere except \(x=1\)

f1 <- function(x){ 1 / (1 - x)}

taylor(f1, var="x", order=4)
## $f
## [1] "(1) * 1 + (1.00000000000064) * x^1 + (0.999999999867712) * x^2 + (0.999994567795319) * x^3 + (0.999693345466748) * x^4"
## 
## $order
## [1] 4
## 
## $terms
##   var      coef degree
## 0   1 1.0000000      0
## 1 x^1 1.0000000      1
## 2 x^2 1.0000000      2
## 3 x^3 0.9999946      3
## 4 x^4 0.9996933      4

Question 2

\[\begin{aligned} f(x) = e^x \end{aligned}\]

This function is valid everywhere within \((-\infty, \infty)\). We can calulate the Taylor series to n=4

# f1 <- function(x){ 1 / (1 - x)}

taylor("exp(x)", var="x", order=4)
## $f
## [1] "(1) * 1 + (1) * x^1 + (0.5) * x^2 + (0.166666666666667) * x^3 + (0.0416666666666667) * x^4"
## 
## $order
## [1] 4
## 
## $terms
##   var       coef degree
## 0   1 1.00000000      0
## 1 x^1 1.00000000      1
## 2 x^2 0.50000000      2
## 3 x^3 0.16666667      3
## 4 x^4 0.04166667      4

Question 3

\[\begin{aligned} f(x) = \log(1 + x) \end{aligned}\]

This function is only defined when \(x > -1\). We can compute the Taylor series, centered at 0

f3 <- function(x){log(1 + x)}

(t3 <- taylor(f3, var=0, order=4))
## $f
## [1] "(0.999999999994524) * x1^1 + (-0.499999999988492) * x1^2 + (0.333332558064607) * x1^3 + (-0.249961920878393) * x1^4"
## 
## $order
## [1] 4
## 
## $terms
##    var       coef degree
## 0    1  0.0000000      0
## 1 x1^1  1.0000000      1
## 2 x1^2 -0.5000000      2
## 3 x1^3  0.3333326      3
## 4 x1^4 -0.2499619      4

Question 4

\[\begin{aligned} f(x) = x^{1/2} = \sqrt{x} \end{aligned}\]

The above function is only valid and differentiable over the reals when \(x > 0\). We can calculate the Taylor series centered at \(x=1\), as thatโ€™s a valid point of reference

f4 <- function(x){sqrt(x)}

(t4 <- taylor(f4, var=1, order=4))
## $f
## [1] "(1) * 1 + (0.499999999993442) * (x1-1)^1 + (-0.125000000007528) * (x1-1)^2 + (0.0624999125751056) * (x1-1)^3 + (-0.0390585237804083) * (x1-1)^4"
## 
## $order
## [1] 4
## 
## $terms
##        var        coef degree
## 0        1  1.00000000      0
## 1 (x1-1)^1  0.50000000      1
## 2 (x1-1)^2 -0.12500000      2
## 3 (x1-1)^3  0.06249991      3
## 4 (x1-1)^4 -0.03905852      4