What Is Binomial Expansion of (1 + x)^n?
For a whole number n, (1 + x)ⁿ expands into n + 1 terms with binomial coefficients as multipliers. For any other n, the expansion is an infinite series that converges only when the absolute value of x is less than 1.
Enter n and the number of terms. The calculator lists each coefficient, writes the expansion and compares a partial sum with the exact value of (1 + x)ⁿ at an x you choose.
The general term is n(n − 1)…(n − k + 1) ÷ k! times xᵏ. For whole n the coefficients stop at k = n, which is why the series ends.
How to Use the Binomial Expansion of (1 + x)^n Calculator
- Enter n.
- Optionally enter x to test.
- Read the coefficients and terms.
- Compare the partial sum with the exact value.
Binomial Expansion of (1 + x)^n Formula
Whole n
General n
Binomial Expansion of (1 + x)^n Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | The power | |
| x | The variable | |
| C(n, k) | Binomial coefficient | |
| |x| < 1 | Convergence condition when n is not a whole number |
Binomial Expansion of (1 + x)^n Worked Example
(1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴. For n = −1 the series is 1 − x + x² − x³ + …, valid for |x| < 1. For n = 1/2 it begins 1 + x/2 − x²/8 + x³/16 − …
With the example values loaded in the calculator above, the steps are:
- For a whole number n the expansion is finite: (1 + x)ⁿ = Σ C(n, k) xᵏ for k = 0 to 6.
- Coefficients C(6, k): 1, 6, 15, 20, 15, 6, 1.
- (1 + 0.1)^6 = 1.771561.
Result: (1 + x)⁶ = 1 + 6x + 15x² + 20x³ + 15x⁴ + 6x⁵ + x⁶; Value at x = 0.1 = 1.771561.
How to Read the Expansion
For a whole number n, the expansion stops. (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴, and the coefficients sum to 2⁴ = 16 because setting x = 1 gives (1 + 1)⁴.
For any other n the expansion never stops, and it is only valid when |x| < 1. For n = −1 it gives 1 − x + x² − x³ + …, and for n = 1/2 it begins 1 + x/2 − x²/8 + x³/16 − …. The partial sum comes close to the exact value for small x and drifts away as x nears 1.
Where Binomial Expansion of (1 + x)^n Is Used
Expanding powers
Expand (1 + x)ⁿ for whole n and read the coefficients from Pascal’s triangle.
Approximating roots
Use the first few terms of the series for n = 1/2 to estimate a square root when x is small.
Checking convergence
Compare a partial sum with the exact value at a chosen x to see how many terms you need.
Calculus practice
Check coefficients for fractional and negative powers term by term.
Common Binomial Expansion of (1 + x)^n Mistakes to Avoid
- For non-whole n the series needs |x| < 1. Outside that range the partial sums do not settle.
- The general coefficient is n(n − 1)…(n − k + 1) ÷ k!. For whole n it becomes zero after k = n, which is why the series stops.
- Use more terms when x is close to 1. Convergence slows down there.
- For (a + x)ⁿ, factor out a first: (a + x)ⁿ = aⁿ(1 + x/a)ⁿ.
Binomial Expansion of (1 + x)^n Terms and Definitions
- Binomial coefficient
- C(n, k), the number of ways to choose k items from n.
Binomial Expansion of (1 + x)^n FAQs
What is the binomial expansion of (1 + x)^n?
It is the sum of C(n, k)·xᵏ for k from 0 up to n when n is a whole number. For n = 4 that gives 1 + 4x + 6x² + 4x³ + x⁴.
How do I expand (1 + x)^n for fractional n?
Use the generalised coefficients n(n − 1)…(n − k + 1) ÷ k!. For n = 1/2 the series is 1 + x/2 − x²/8 + x³/16 − …, and it goes on forever.
When is the series valid?
For a whole n, always. For any other n, the series converges only when |x| < 1.
What do the coefficients add to?
For whole n they add to 2ⁿ, found by setting x = 1. For n = 6 the sum is 64.