Theory of Intelligence
11TH•Mathematics•Mathematical Inductions And Binomial Theorem•8.3 Binomial Series

8.3 Binomial Theorem (Negative or Fractional Index)

Binomial expansion for negative or fractional index when |x| < 1.

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8.3 Binomial Theorem (Negative or Fractional Index)

For ∣x∣<1|x| < 1 and nn not necessarily a positive integer:

(1+x)n=1+nx+n(n−1)2!x2+⋯+n(n−1)⋯(n−r+1)r!xr+⋯(1+x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \cdots + \dfrac{n(n-1)\cdots(n-r+1)}{r!}x^r + \cdots

General term: Tr+1=n(n−1)⋯(n−r+1)r!xrT_{r+1} = \dfrac{n(n-1)\cdots(n-r+1)}{r!}x^r.

Example 18

For (1+x)−3(1+x)^{-3}: Tr+1=(−1)r(r+1)(r+2)2xrT_{r+1} = (-1)^r \dfrac{(r+1)(r+2)}{2}x^r.

Particular cases:

(1+x)−1=1−x+x2−x3+⋯(1+x)−2=1−2x+3x2−4x3+⋯(1−x)−1=1+x+x2+x3+⋯(1−x)−2=1+2x+3x2+⋯\begin{align*} (1+x)^{-1} &= 1 - x + x^2 - x^3 + \cdots \\ (1+x)^{-2} &= 1 - 2x + 3x^2 - 4x^3 + \cdots \\ (1-x)^{-1} &= 1 + x + x^2 + x^3 + \cdots \\ (1-x)^{-2} &= 1 + 2x + 3x^2 + \cdots \end{align*}

Example 19

Coefficient of xnx^n in 1−x(1+x)2\dfrac{1-x}{(1+x)^2} is (−1)n(2n+1)(-1)^n(2n+1).

Example 20

For small xx (neglect x3+x^3+): 1−x1+x≈1−x+12x2\sqrt{\dfrac{1-x}{1+x}} \approx 1 - x + \dfrac{1}{2}x^2.

Example 21

Given series for yy, show 5y2+10y−4=05y^2 + 10y - 4 = 0 by matching (1+x)n(1+x)^n with n=−12n = -\dfrac{1}{2}, x=−49x = -\dfrac{4}{9}.

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8.2 Binomial Theorem
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