Question

Find the first 4 terms of the binomial series for the function f(x) = (1 + x/3)^-1/3, 1st term: 2nd term: 3rd term: 4th term:

Answer 1

To find the first 4 terms of the binomial series for the function f(x) = (1 + x/3)^(-1/3), we can use the binomial theorem. The first term is 1, the second term is -x/9, the third term is x^2/81, and the fourth term is -x^3/729.

Step-by-step explanation:

The binomial series expansion for the function f(x) = (1 + x/3)^(-1/3) can be obtained using the binomial theorem. The binomial theorem states that for any real numbers a and b and any positive integer n, the expansion of (a + b)^n can be given by:

(a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n)a^0 b^n

Where C(n,k) represents the binomial coefficient n choose k.

In this case, we have a = 1, b = x/3, and n = -1/3. Therefore, we can calculate the first four terms of the binomial series as follows:

1. 1st term: C(-1/3, 0) (1)^(-1/3) (x/3)^0 = 1/1 = 1
2. 2nd term: C(-1/3, 1) (1)^(-4/3) (x/3)^1 = -1/3 (x/3) = -x/9
3. 3rd term: C(-1/3, 2) (1)^(-7/3) (x/3)^2 = 1/9 (x/3)^2 = x^2/81
4. 4th term: C(-1/3, 3) (1)^(-10/3) (x/3)^3 = -1/27 (x/3)^3 = -x^3/729

Answer 2

The first 4 terms of the binomial series for
`f(x) = (1 + x/3)^-1/3` are provided using the binomial theorem.

Step-by-step explanation:

The binomial series is a mathematical representation of the expansion of a binomial expression raised to a positive integer power. It provides a formula for efficiently calculating the coefficients of the expansion.

The binomial series for the function
`f(x) = (1 + x/3)^-1/3` can be found using the binomial theorem. The first 4 terms of the binomial series are:

1. 1st term:
   `(1 + x/3)^-1/3`
2. 2nd term:
   `(-1/3)(x/3)(1 + x/3)^-4/3`
3. 3rd term:
   `(-1/3)(-4/3)(x/3)^2(1 + x/3)^-7/3`
4. 4th term:
   `(-1/3)(-4/3)(-7/3)(x/3)^3(1 + x/3)^-10/3`