Binomial Series Expansion Calculator
Enter n as a fraction, negative number or whole number to expand (1 + x)^n or (a + bx)^n term by term with exact coefficients, see the general term and each step, the values of x for which the series is valid, and how close the first terms get to the true value.
The binomial series is (1 + x)^n = 1 + nx + n(n − 1)/2!·x² + n(n − 1)(n − 2)/3!·x³ + …, with general term C(n, k)·x^k where C(n, k) = n(n − 1)…(n − k + 1)/k!.
Use the Binomial Series Expansion Calculator
Binomial series (a + bx)n — any n
n may be a fraction (1/2, −2/3), a decimal or a whole number. Coefficients are exact fractions.
| k | C(n, k) = n(n−1)…(n−k+1)/k! | × (b/a)k | Term |
|---|
Finite expansion (a + b)n for whole n (0–10)
( a + b ) n
Expanded Expression
Term Breakdown
| k | C(n,k) | Term (symbolic) | Numeric value |
|---|
Pascal's Triangle Row (n = )
Set a, b, and n then click Expand
Common binomial series expansions
Infinite series for negative and fractional powers. Replace x with bx/a and multiply by an for (a + bx)n.
| Expression | Expansion | General term | Valid for |
|---|---|---|---|
| (1 + x)^−1 | 1 − x + x² − x³ + x⁴ − … | (−1)^k x^k | |x| < 1 |
| (1 − x)^−1 | 1 + x + x² + x³ + x⁴ + … | x^k | |x| < 1 |
| (1 + x)^−2 | 1 − 2x + 3x² − 4x³ + 5x⁴ − … | (−1)^k (k + 1) x^k | |x| < 1 |
| (1 + x)^½ | 1 + x/2 − x²/8 + x³/16 − 5x⁴/128 + … | C(½, k) x^k | |x| < 1 |
| (1 + x)^−½ | 1 − x/2 + 3x²/8 − 5x³/16 + 35x⁴/128 − … | C(−½, k) x^k | |x| < 1 |
| (1 − x)^−½ | 1 + x/2 + 3x²/8 + 5x³/16 + 35x⁴/128 + … | C(2k, k) x^k / 4^k | |x| < 1 |
| (1 + x)^⅓ | 1 + x/3 − x²/9 + 5x³/81 − 10x⁴/243 + … | C(⅓, k) x^k | |x| < 1 |
| (1 + x)^(3/2) | 1 + 3x/2 + 3x²/8 − x³/16 + 3x⁴/128 − … | C(3/2, k) x^k | |x| < 1 |
Quick answer
The binomial series is (1 + x)^n = 1 + nx + n(n − 1)/2!·x² + n(n − 1)(n − 2)/3!·x³ + …, with general term C(n, k)·x^k where C(n, k) = n(n − 1)…(n − k + 1)/k!. For a whole number n ≥ 0 it stops after n + 1 terms; for a negative or fractional n it is infinite and valid only for |x| < 1. For (a + bx)^n, factor out a^n: (a + bx)^n = a^n·(1 + bx/a)^n, valid for |bx/a| < 1. Example: √(1 + x) = (1 + x)^(1/2) ≈ 1 + x/2 − x²/8 + x³/16, so √1.1 ≈ 1.0488125 against the true 1.0488088.
Examples
| Expression | Series (first four terms) | Valid for |
|---|---|---|
| (1 + x)^−1 | 1 − x + x² − x³ + … | |x| < 1 |
| (1 + x)^−2 | 1 − 2x + 3x² − 4x³ + … | |x| < 1 |
| (1 + x)^½ | 1 + x/2 − x²/8 + x³/16 − … | |x| < 1 |
| (1 + x)^−½ | 1 − x/2 + 3x²/8 − 5x³/16 + … | |x| < 1 |
| (1 − 2x)^−1 | 1 + 2x + 4x² + 8x³ + … | |x| < ½ |
| (4 + x)^½ | 2 + x/4 − x²/64 + x³/512 − … | |x| < 4 |
Type n as a fraction such as -1/2 or 1/3, choose a and b, and set x to see the step-by-step expansion and the approximation error.
Summary
Newton's binomial series expands (1 + x)^n for any real n. This calculator handles fractional and negative powers as well as whole ones: it factors (a + bx)^n into a^n(1 + bx/a)^n, gives every coefficient as an exact fraction, shows the general term and the working for each term, states the interval |x| < |a/b| on which the infinite series converges, and compares the partial sum with the exact value at your x. A second tool expands (a + b)^n for whole n from 0 to 10 with Pascal's triangle coefficients.
How it works
- Enter a, b and n for (a + bx)^n. n can be a whole number, a negative number, a decimal or a fraction such as -1/2 or 2/3.
- Choose how many terms you want and an x value to test the approximation.
- The tool rewrites (a + bx)^n as a^n·(1 + bx/a)^n and builds each coefficient C(n, k)·(b/a)^k as an exact fraction from n(n − 1)…(n − k + 1)/k!.
- Read the expansion, the general term, the step-by-step table and the range of x for which the infinite series converges, |bx/a| < 1.
- Compare the partial sum with the exact value of (a + bx)^n at your x to see the error. For whole-number powers of (a + b), use the finite expansion below.
Use cases
- A-level and calculus homework: expand (1 + x)^n in ascending powers of x up to x³ and state the validity.
- Approximating roots and reciprocals, for example √1.02 or 1/0.98, by hand.
- Checking expansions of (a + bx)^n where a ≠ 1, such as (4 + x)^(1/2) or (2 − 3x)^(−2).
- Seeing how many terms are needed for a given accuracy.
- Expanding (a + b)^n for whole n with Pascal's triangle coefficients.