Calculus How To

Binomial Series

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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 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) = a2 + 2ab + b2

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:
binomial series example

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)

Step 1Calculate 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:
function to binomial series


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.
solution for binomial expansion

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.

CITE THIS AS:
Stephanie Glen. "Binomial Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/binomial-series/
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