Calculus of a Single Variable, Chapter 9, 9.10, Section 9.10, Problem 19

Binomial series is an example of an infinite series. When it is convergent at |x|lt1 , we may follow the sum of the binomial series as (1+x)^k where k is any number. We may follow the formula:
(1+x)^k = sum_(n=0)^oo (k(k-1)(k-2) ...(k-n+1))/(n!) x^n
or
(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+...
To evaluate the given function f(x) = 1/sqrt(1-x) , we may apply radical property: sqrt(x) = x^(1/2) . The function becomes:
f(x) = 1/ (1-x)^(1/2)
Apply Law of Exponents: 1/x^n = x^(-n) to rewrite the function as:
f(x) = (1-x)^(-1/2)
or f(x)= (1 -x)^(-0.5)
This now resembles (1+x)^k form. By comparing "(1+x)^k " with "(1 -x)^(-0.5) or (1+(-x))^(-0.5) ”, we have the corresponding values:
x=-x and k =-0.5 .
Plug-in the values on the aforementioned formula for the binomial series, we get:
(1-x)^(-0.5) =sum_(n=0)^oo (-0.5(-0.5-1)(-0.5-2)...(-0.5-n+1))/(n!)(-x)^n
=1 + (-0.5)(-x) + (-0.5(-0.5-1))/(2!) (-x)^2 + (-0.5(-0.5-1)(-0.5-2))/(3!)(-x)^3 +(-0.5(-0.5-1)(-0.5-2)(-0.5-3))/(4!)(-x)^4+...
=1 + 0.5x + (-0.5(-1.5))/(1*2) (-1)^2x^2 + (-0.5(-1.5)(-2.5))/(1*2*3) (-1)^3x^3 +(-0.5(-1.5)(-2.5)(-3.5))/(1*2*3*4)(-1)^4x^4+...
=1 + 0.5x + 0.75/2 (1)x^2 + (-1.875)/6 (-1)x^3 +(6.5625)/24(1)x^4+...
=1 + 1/2x + (3x^2)/8 + (5x^3)/16 +(35x^4)/128+...
Therefore, the Maclaurin series for the function f(x) =1/sqrt(1-x) can be expressed as:
1/sqrt(1-x)=1 + x/2 + (3x^2)/8 + (5x^3)/16 +(35x^4)/128+...