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Foundations: Maths

Binomial expansion (first three terms)

Approximate (1 + x)ᵏ for small x.

Ref: FMA-20
Rev: v1
Field: Foundations: Maths
Result: (1+x)ᵏ

{(1 + x)}^{k} \approx 1 + k x + \frac{k (k - 1)}{2} x^{2}

For non-integer k the series holds for |x| < 1. For a non-negative integer k it terminates and holds for every x. The calculator uses the first three terms.

For small x, a power of 1 + x is well approximated by the first few terms of its binomial series.

Calculator

Solve for

(1+x)ᵏApproximationComputed

kPower k

0.53

xSmall quantity x

0.010.5

Exact (1 + x)ᵏ1.049
Error of approximation−0.005611 %

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Approximation () current value 1.049

kPower kKey driver

1.33

xSmall quantity x

11.22

Square roots, reciprocals and small fractional changes can be approximated without a calculator.
Keeping only the first two terms gives the common linear approximation (1+x)ᵏ ≈ 1 + kx.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
k Power k	0.5 to 3	1.049	1.33	0.2813	+undefined%	—
x Small quantity x	0.01 to 0.5	1.005	1.219	0.2138	+undefined%	—

Sweep of Small quantity x

Small quantity x	Approximation
0.01	1.005
0.07125	1.035
0.1325	1.064
0.1938	1.092
0.255	1.119
0.3163	1.146
0.3775	1.171
0.4387	1.195
0.5	1.219

Sweep one variable

Vary

Approximation ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

(1+x)ᵏ

Approximation

Result ·

What the equation gives you. Everything else feeds it.

Power k

· typical 0.5 to 3

Index of the bracket. Larger k magnifies the effect of x.

Small quantity x

· typical 0.01 to 0.5

The small term; the series needs |x| below 1. The approximation improves as x gets smaller.

Worked example with the values above

Linear term: kx = 0.5 × 0.1 = 0.05.
Quadratic term: k(k−1)x²/2 = −0.00125.
(1+x)^k ≈ 1 + 0.05 + −0.00125 = 1.04875.

Assumptions and limits

|x| is less than 1 for non-integer k.
Only three terms are kept.

Common mistakes

Using x ≥ 1 for non-integer k.
Forgetting the factor ½ in the third term.

Statistics and uncertainty

The truncation error is about the size of the first omitted term, of order x³; halving x reduces the error roughly eightfold.

Where it is used

Small-change estimates in engineering formulas.
Linearising relationships near an operating point.

Maths you need

Powers: fractional and negative indices
Factorials: binomial coefficients

Related formulas

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