Taylor Series Approximations Discussion 14

8.8 Excercise 21-24

In Exercises 21 – 24, write out the first 5 terms of the Binomial series with the givenk-value. 21.k=½ 22.k= -1/2 23.k=1/3 24.k=4

Binomial Series : (1+x)^k = 1 + kx + (k(k-1)/2!) x^2 + (k(k-1)(k-2)/3!) x^3 + (k(k-1)(k-2)(k-3)/4!) x^4 + …

x <-2
#Binomial

k = 1/2

# Find the first 5 terms of the Binomial series for k = 1/2
k <- 1/2
term <- 1+ k * x + (k * (k-1) / 2) * x^2 +  (k * (k-1) * (k-2) / 6) * x^3 + (k * (k-1) * (k-2) * (k-3) / 24) * x^4

k = -1/2

# Find the first 5 terms of the Binomial series for k = -1/2
k <- -1/2
term <- 1+ k * x + (k * (k-1) / 2) * x^2 +  (k * (k-1) * (k-2) / 6) * x^3 + (k * (k-1) * (k-2) * (k-3) / 24) * x^4

k = 1/3

# Find the first 5 terms of the Binomial series for k = 1/3
k <- 1/3
term <- 1+ k * x + (k * (k-1) / 2) * x^2 +  (k * (k-1) * (k-2) / 6) * x^3 + (k * (k-1) * (k-2) * (k-3) / 24) * x^4

k = 4

# Find the first 5 terms of the Binomial series for k = 4
k <- 4
term <- 1+ k * x + (k * (k-1) / 2) * x^2 +  (k * (k-1) * (k-2) / 6) * x^3 + (k * (k-1) * (k-2) * (k-3) / 24) * x^4