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The binomial series for any rational n

The binomial expansion from P2 also works when n is negative or a fraction. The series then goes on for ever and is only valid for some values of x.

The expansion

(1 + x)n = 1 + nx + n(n - 1)/2! x2 + n(n - 1)(n - 2)/3! x3 + ...

For a positive whole number n it stops. For other rational n it is an infinite series.

Validity

For n not a positive integer, the series is valid only for |x| < 1. For (1 + kx)n replace x by kx: it is valid for |kx| < 1, so |x| < 1/|k|. Always put kx in brackets: (1 + 2x)-1 uses (2x)2 = 4x2.

Approximations

For small x, a few terms give a good approximation. To estimate sqrt(1.02), use (1 + x)1/2 with x = 0.02, which is inside the valid range.

Worked example

Expand (1 + x)-2 up to the x3 term.

  1. n = -2: nx = -2x
  2. n(n - 1)/2 = (-2)(-3)/2 = 3
  3. n(n - 1)(n - 2)/6 = (-2)(-3)(-4)/6 = -4

Answer: 1 - 2x + 3x2 - 4x3 + ..., valid for |x| < 1

Key idea: (1 + x)n = 1 + nx + n(n - 1)x2/2! + ...; for non-integer or negative n it is infinite and valid for |x| < 1. Bracket kx.

Check yourself

  1. Find the coefficient of x2 in the expansion of (1 + x)1/2. Give your answer as a fraction in its simplest form.
    Show answer

    -1/8. (1/2)(-1/2)/2 = -1/8.

  2. In the binomial series for (1 - 2x)-1/2, what is the x3 coefficient? Give your answer as a fraction in its simplest form.
    Show answer

    5/2. (-1/2)(-3/2)(-5/2)/3! = -5/16. Multiply by (-2)3 = -8: (-5/16)(-8) = 5/2.

  3. For which values of x is the expansion of (1 + 3x)-1 valid?
    • |x| < 1
    • |x| < 1/3
    • |x| < 3
    • x > -1/3
    Show answer

    |x| < 1/3. It is valid for |3x| < 1, so |x| < 1/3.

  4. Find the coefficient of x in the expansion of (1 + 4x)3/2.
    Show answer

    6. nkx = (3/2)(4x) = 6x.

  5. Use the first three terms of the expansion of (1 + x)1/2 with x = 0.02 to estimate sqrt(1.02), to 5 decimal places.
    Show answer

    1.00995. 1 + (1/2)(0.02) - (1/8)(0.02)2 = 1 + 0.01 - 0.00005 = 1.00995.

  6. Which is the expansion of (1 - x)-1?
    • 1 - x + x2 - x3 + ...
    • 1 - x
    • 1 + x + x2 + x3 + ...
    • 1 + x
    Show answer

    1 + x + x2 + x3 + .... With n = -1, the term in yr of (1 + y)-1 is (-1)r yr. Put y = -x: (-1)r(-x)r = xr, so every coefficient is 1.

  7. Expand (1 + x)-3 as far as the x2 term. What number multiplies x2?
    Show answer

    6. (-3)(-4)/2 = 6.

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Last updated 7 November 2026. Spotted a mistake? Tell us.