The **binomial series** is a type of Maclaurin series for the power function f(x) = (1 + x)^{m}. You can find the series expansion with a formula:

## Binomial Series vs. Binomial Expansion

The “binomial series” is named because it’s a series—the sum of terms in a sequence (for example, 1 + 2 + 3) and it’s a “binomial”— two quantities (from the Latin *binomius*, which means “two names”). The two terms are enclosed within parentheses. For example (a + b) and (1 + x) are both binomials. When these quantities are raised to power and expanded, we get a **binomial expansion**:

- (a + b)
^{0}= 1 - (a + b)
^{1}= a + b - (a + b)
^{2}= (a + b) * (a + b) = a^{2}+ 2ab + b^{2}

**Once you get above the fourth power, the algebra becomes tedious.** You don’t have to calculate these out completely though: there’s a shortcut of sorts. The formula gives the **expansion of any binomial series**, but you’ll still have to work through some algebra to actually expand it.

## Expressing a Function as a Binomial Series

The formula for expanding a binomial series can also be used to simplify more complex functions. The Σ in the formula is summation notation, which basically means to “add everything up”. The (m k) is the binomial coefficient, equal to m! / (k! (m – k)!), where the ! symbol is a factorial.

**Example question:** Express the following function as a binomial series:

**Solution**:

Note that the square root in the denominator can be rewritten with algebra as a power (to -½), so we can use the formula with the rewritten function (1 + x)^{-½}

Step 1**Calculate the first few values for the binomial coefficient (m k).** What you’re looking for here is a *pattern* for some arbitrary value for “k”. So, you’ll have to work the algebra until you can clearly see a pattern. The first two values for the expansion are:

- -½!/(0!(-½-0)!) = 1
- -½!/(1!(-½-1)!) = -½

These don’t give much of a clue, so let’s continue to the third and fourth terms:

A pattern is emerging, so we can generalize the expansion for any “k”, to:

**Note**: If you don’t see the pattern, continue with finding coefficients until you do! A pattern will nearly always emerge after the third of fourth binomial coefficient, so if you don’t have a pattern by then—go back and check your algebra.

Step 2: Write the solution.

All you’re doing here is writing out the terms you calculated in Step 1 (shown in the yellow boxes), followed by the corresponding power of x (blue boxes).

## Convergence of The Binomial Series

The ratio test can be used to show that the series converges for absolute values of x less than 1, |x| < 1 (to the expected sum (1 + x)^{k}) and diverges for |x| > 1. In addition, the radius of convergence is R = 1, unless the exponent (k) is a positive whole number.

**Fun Fact:** the binomial series formula is inscribed on Newton’s gravestone (at his request) at Westminster Abbey (Nitecki, p. 367).

**Next**: Gauss Hypergeometric Function (a generalization of the binomial series).

## References

Bolton, W. (2000). Mathematics for Engineering. Newness.

Farahmand, A. 11.10 More About Taylor Series; Binomial Series. Retrieved September 22, 2020 from: https://math.berkeley.edu/~arash/notes/07_01.pdf

Gonzalez-Zugast, J. (2011). The Binomial Series. Retrieved September 23, 2020 from: http://faculty.uml.edu/jennifer_gonzalezzugasti/Calculus%20II%20Video%20Lectures/documents/9.10.1TheBinomialSeries.pdf

Nitecki, Z. (2009). Calculus Deconstructed. A Second Course in First-Year Calculus.

Olive, J. (2003). Maths: A Student’s Survival Guide: A Self-Help Workbook for Science and Engineering Students. Cambridge University Press.

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