HW14

1

f <- function(x) 1/(1-x)
f_prime <- function(x) 1/(1-x)^2
f_double_prime <- function(x) 2/(1-x)^3
f_triple_prime <- function(x) 6/(1-x)^4

f_prime(0) # 1

## [1] 1

f_double_prime(0) # 2

## [1] 2

f_triple_prime(0) # 6

## [1] 6

f_n <- function(n, x) factorial(n)*(1-x)^(n+1)
sum_n <- function(x) sum(sapply(0:Inf, function(n) x^n/factorial(n)))

2

f <- function(x) exp(x)
f_n <- function(n, x) exp(x)
en_0 <- 1
sum_n <- function(x) sum(sapply(0:Inf, function(n) x^n/factorial(n)))

3

f <- function(x) log(1+x)
f_prime <- function(x) 1/(1+x)
f_double_prime <- function(x) -1/(1+x)^2
f_triple_prime <- function(x) 2/(1+x)^3
f_quad_prime <- function(x) -6/(1+x)^4

f_prime(0) # 1

## [1] 1

f_double_prime(0) # -1

## [1] -1

f_triple_prime(0) # 2

## [1] 2

f_quad_prime(0) # -6

## [1] -6

sum_n <- function(x) sum(sapply(1:Inf, function(n) (-1)^(n+1)*x^n/n))

4

f <- function(x) {x^(1/2)}