write down the first four terms of the following expansion (1+x)^1/4 stating in each case, the values of x for which the expansions are valid
To avoid writing all the powers of 1, let's agree that they are all just 1. So,
(1+x)^(1/4)
= 1 + (1/4)x^1 + (1/4)(-3/4)/2! x^2 + (1/4)(-3/4)(-7/4)/3! x^3 + ...
= 1 + x/4 - 3x^2/32 + 7x^3/128 - ...
To expand the binomial (1+x)^(1/4), we can use the binomial theorem. The binomial theorem states that:
(1+x)^n = C(n,0) * 1^(n-0) * x^0 + C(n,1) * 1^(n-1) * x^1 + C(n,2) * 1^(n-2) * x^2 + ...
Where C(n,k) represents the binomial coefficient. In this case, n = 1/4.
The binomial coefficient C(n,k) is given by:
C(n,k) = n! / (k! * (n-k)!)
Let's calculate the first four terms:
Term 1:
C(1/4,0) * 1^(1/4) * x^0
= 1 * x^0
= 1
Term 2:
C(1/4,1) * 1^(1/4-1) * x^1
= (1/4) * 1 * x
= 1/4 * x
Term 3:
C(1/4,2) * 1^(1/4-2) * x^2
= (1/4 * (1/4 - 1)) * 1 * x^2
= (1/4 * (-3/4)) * x^2
= -3/16 * x^2
Term 4:
C(1/4,3) * 1^(1/4-3) * x^3
= (1/4 * (1/4 - 1) * (1/4 - 2)) * 1 * x^3
= (1/4 * (-3/4) * (-5/4)) * x^3
= 15/64 * x^3
The first four terms of the expansion are:
Term 1: 1
Term 2: 1/4 * x
Term 3: -3/16 * x^2
Term 4: 15/64 * x^3
For the expansion (1+x)^(1/4) to be valid, x must satisfy the condition |x| < 1, meaning that the absolute value of x should be less than 1.
To find the first four terms of the expansion of (1+x)^(1/4), we can use the binomial theorem. The binomial theorem allows us to expand expressions of the form (a+b)^n, where "a" and "b" are any real numbers and "n" is a positive integer.
The formula for the binomial theorem is as follows:
(a+b)^n = C(n,0) * a^(n-0) * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,n-1) * a^1 * b^(n-1) + C(n,n) * a^0 * b^n
In our case, a = 1, b = x, and n = 1/4. We want to find the first four terms, so we need to calculate the coefficients and exponents for each term.
To determine the values of x for which the expansions are valid, we need to consider the convergence of the series. In this case, the expansion is valid when the absolute value of x is less than 1. Therefore, |x| < 1.
Let's calculate the first four terms:
term 1: C(1/4,0) * 1^(1/4-0) * x^0 = 1
term 2: C(1/4,1) * 1^(1/4-1) * x^1 = (1/4) * x
term 3: C(1/4,2) * 1^(1/4-2) * x^2 = (1/4) * (-3/4) * x^2
term 4: C(1/4,3) * 1^(1/4-3) * x^3 = (1/4) * (-3/4) * (7/4) * x^3
Therefore, the first four terms of the expansion are:
1 - (1/4)x - (3/32)x^2 + (7/256)x^3
And the values of x for which the expansions are valid are:
|x| < 1