binomial series: meaning, definition, pronunciation and examples
C2Technical / Academic (Mathematics)
Quick answer
What does “binomial series” mean?
The infinite series expansion of a binomial expression (1+x)^n, valid for |x|<1, regardless of whether n is an integer.
Audio
Pronunciation
Definition
Meaning and Definition
The infinite series expansion of a binomial expression (1+x)^n, valid for |x|<1, regardless of whether n is an integer.
A fundamental power series in calculus and analysis derived from the binomial theorem, used for approximations and in deriving other series like the exponential series.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. Potential minor variation in pronunciation of 'binomial' and in phrasing (e.g., 'expand using the binomial series' vs. 'apply the binomial series expansion').
Connotations
Identical technical meaning in both varieties.
Frequency
Used with identical, high frequency in university-level mathematics contexts in both regions.
Grammar
How to Use “binomial series” in a Sentence
the binomial series for (1+x)^nexpand (expression) as a binomial seriesthe binomial series converges for |x| < 1Vocabulary
Collocations
Examples
Examples of “binomial series” in a Sentence
adjective
British English
- The binomial-series expansion is convergent in the unit disk.
American English
- The binomial-series expansion converges within the unit disk.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core concept in calculus, real analysis, and mathematical methods for physics/engineering.
Everyday
Not used in everyday conversation.
Technical
Precise term used in mathematical derivations, approximations (e.g., in engineering or physics calculations), and computational algorithms.
Vocabulary
Synonyms of “binomial series”
Strong
Neutral
Weak
Vocabulary
Antonyms of “binomial series”
Watch out
Common Mistakes When Using “binomial series”
- Using it to refer to the finite expansion for integer n.
- Misapplying it for |x| >= 1 without discussing convergence.
- Confusing the binomial series coefficients (generalised binomial coefficients) with standard combinatorial 'n choose k'.
FAQ
Frequently Asked Questions
The binomial theorem gives a finite expansion when the exponent is a positive integer. The binomial series gives an infinite series expansion valid for any real (or complex) exponent, provided |x| < 1.
It was generalized to non-integer exponents by Sir Isaac Newton in the 1660s.
Yes, the binomial series for (1+x)^n is precisely the Taylor series of that function expanded about x=0.
Yes, by factoring out a^n to get a^n(1 + x/a)^n, then applying the series to the (1 + x/a)^n part.
The infinite series expansion of a binomial expression (1+x)^n, valid for |x|<1, regardless of whether n is an integer.
Binomial series is usually technical / academic (mathematics) in register.
Binomial series: in British English it is pronounced /baɪˌnəʊ.mi.əl ˈsɪə.riːz/, and in American English it is pronounced /baɪˌnoʊ.mi.əl ˈsɪr.iːz/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Bi-nomial' means 'two names' (two terms: '1' and 'x'). The 'series' is the long, infinite list you get when the power 'n' isn't a simple whole number.
Conceptual Metaphor
UNPACKING A COMPACTED POWER: Treating (1+x)^n as a compressed form that can be unpacked into an infinite sum of simpler x^k terms when n is not an integer.
Practice
Quiz
For which of the following values of x is the binomial series for (1+x)^{-2} guaranteed to converge?