Page 496 Exercises 8.8
In Exercises 21 – 24, write out the first 5 terms of the Binomial series with the given k-value.
The binomial coefficient formula for non-integer values of k is given by:
C(k,r) = k(k-1)(k-2)…(k-r+1)/(r!)
If k=1/2, then the first five terms of the Binomial series are:
(1+x)^(1/2) = C(1/2,0) + C(1/2,1)x + C(1/2,2)x^2 + C(1/2,3)x^3 + C(1/2,4)x^4 = 1 + (1/2)x - (1/8)x^2 + (1/16)x^3 - (5/128)x^4
Therefore, the first five terms of the Binomial series with k=1/2 are 1, (1/2)x, -(1/8)x^2, (1/16)x^3, and -(5/128)x^4.
The Binomial series can be defined using the following formula for the binomial coefficient:
C(k,r) = (-1)^r * (-k+r-1)! / (r! * (-k)!)
If k=-1/2, then the first five terms of the Binomial series are:
(1+x)^(-1/2) = C(-1/2,0) + C(-1/2,1)x + C(-1/2,2)x^2 + C(-1/2,3)x^3 + C(-1/2,4)x^4 = 1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4
Therefore, the first five terms of the Binomial series with k=-1/2 are 1, -(1/2)x, (3/8)x^2, -(5/16)x^3, and (35/128)x^4.
The Binomial series can be defined using the following formula for the binomial coefficient:
C(k,r) = k(k-1)(k-2)…(k-r+1)/(r!)
If k=1/3, then the first five terms of the Binomial series are:
(1+x)^(1/3) = C(1/3,0) + C(1/3,1)x + C(1/3,2)x^2 + C(1/3,3)x^3 + C(1/3,4)x^4 = 1 + (1/3)x - (2/27)x^2 + (4/81)x^3 - (26/729)x^4
Therefore, the first five terms of the Binomial series with k=1/3 are 1, (1/3)x, -(2/27)x^2, (4/81)x^3, and -(26/729)x^4.
The Binomial series is given by:
(1+x)^k = C(k,0) + C(k,1)x + C(k,2)x^2 + C(k,3)x^3 + …
where C(k,r) denotes the binomial coefficient “k choose r”, which is equal to k!/(r!(k-r)!).
If k=4, then the first five terms of the Binomial series are:
(1+x)^4 = C(4,0) + C(4,1)x + C(4,2)x^2 + C(4,3)x^3 + C(4,4)x^4 = 1 + 4x + 6x^2 + 4x^3 + x^4
Therefore, the first five terms of the Binomial series with k=4 are 1, 4x, 6x^2, 4x^3, and x^4.