Binomial Series Expander

Math Algebra • Sequences and Series

Written by STEM Calculators Team Published December 19, 2025 Updated February 24, 2026

5. Binomial Series Expander

Expand \((1 + s x)^n\) for integer or fractional/real \(n\). Shows generalized binomial coefficients, truncation, convergence/radius note, and a zoomable plot comparing the function vs partial sums.

Function vs partial sum

x-axis: x • y-axis: y. Drag to pan • wheel/pinch to zoom. Partial sum degree is \(M\).

x: 0, y: 0 sx: 60, sy: 35 Function \((1+s x)^n\) Partial sum \(P_M(x)\)

Click Expand to generate coefficients and the truncated expansion.

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Theory: The Binomial Theorem and Binomial Series

The binomial family \((1+x)^n\) behaves in two different ways depending on the exponent \(n\). If \(n\) is a nonnegative integer, the expansion is a finite polynomial. If \(n\) is not a nonnegative integer (for example a fraction like \(-\tfrac12\)), the expansion becomes an infinite power series around \(x=0\).

1) Integer exponents (finite binomial theorem)

For \(n\in\mathbb{Z}_{\ge 0}\),

\[ (1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k, \qquad \binom{n}{k}=\frac{n!}{k!(n-k)!}. \]

In this case the expansion is exact for all real \(x\) because it is literally a polynomial.

2) Non-integer exponents (generalized binomial series)

For general real \(n\), define the generalized binomial coefficient:

\[ \binom{n}{k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}, \qquad k\ge 1,\quad \binom{n}{0}=1. \]

Then the binomial series around \(x=0\) is

\[ (1+x)^n=\sum_{k=0}^{\infty}\binom{n}{k}x^k, \qquad \text{valid for }|x|<1. \]

If your expression is \((1+s x)^n\), the convergence condition becomes \(|s x|<1\), i.e. \(|x|<\frac{1}{|s|}\).

3) Truncation and big-O notation

In practice, we often keep only the first few terms:

\[ (1+x)^n \approx \sum_{k=0}^{K}\binom{n}{k}x^k + O\!\left(x^{K+1}\right). \]

The \(O(x^{K+1})\) means “terms of order \(x^{K+1}\) and higher.” The plot shows how the partial sum improves as \(K\) increases.

4) Helpful recurrence for computation

Instead of using factorials, coefficients can be generated efficiently by a recurrence:

\[ \binom{n}{0}=1,\qquad \binom{n}{k}=\binom{n}{k-1}\cdot\frac{n-(k-1)}{k}. \]

This works for both integer and non-integer \(n\), and avoids very large factorials.

5) Solved examples (the same ones as “Fill example”)

Example A: \((1+x)^{-1/2}\) (truncate at \(x^4\))

\[ (1+x)^{-1/2} = 1 - \frac12 x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 + O(x^5), \qquad |x|<1. \]

Example B: \((1+2x)^5\) (exact polynomial)

\[ (1+2x)^5=\sum_{k=0}^{5}\binom{5}{k}(2x)^k = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5. \]

Frequently Asked Questions

What does the binomial series expander do for (1 + s x)^n?

It writes (1 + s x)^n as a binomial polynomial when n is a nonnegative integer, or as a truncated binomial power series when n is not a nonnegative integer. The output lists coefficients and the truncated form up to x^K.

How do generalized binomial coefficients work for fractional or real n?

For k >= 1, the generalized coefficient is C(n, k) = n(n-1)(n-2)...(n-k+1) / k!, with C(n, 0) = 1. These coefficients define the power series terms C(n, k) x^k.

What is the convergence condition for (1 + s x)^n as a series?

The generalized binomial series is valid when |s x| < 1, which means |x| < 1/|s| when s is not zero. Outside that interval, the infinite series does not converge to the function.

Why does the partial sum P_M(x) differ from (1 + s x)^n on the plot?

P_M(x) keeps only the first M terms of the series, so it approximates the function near x = 0 and improves as M increases. Differences become larger when x approaches or exceeds the convergence bound.

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