Binomial Series
Summary: Newton’s “universal theorem” for as an infinite series in powers of — Euler’s §71 principal tool for expanding irrational functions. When and is a polynomial in , the expansion is a recurrent series with a non-constant law (§73–§76).
Sources: chapter4
Last updated: 2026-04-23
The universal theorem (§71)
Euler states Newton’s binomial theorem for arbitrary rational exponent :
The series terminates iff is a positive integer; otherwise it has infinitely many terms (source: chapter4, §71).
Sample cases (§71)
Explicit expansions tabulated by Euler:
Term-to-term recurrence (§72)
From one term to the next: if a term has the form , the next term is
The exponent of decreases by each step; the exponent of increases by . Equivalently, extracting as an overall factor gives , and setting ,
which Euler remarks is the form he will usually use (source: chapter4, §72). Here may be any real number — fractional or integer.
Polynomial : recurrent laws (§73–§76)
When is a polynomial in , the expansion of is itself a series in , and its coefficients satisfy a recurrence whose order equals the number of nonzero coefficients in .
(§73)
Writing the series as , the recurrence is
So each coefficient is determined by the one before (source: chapter4, §73).
(§74)
has coefficients determined from the two preceding ones:
Starting values: , , etc. (source: chapter4, §74).
(§75)
Three-term recurrence:
General (§76)
has each coefficient determined by as many predecessors as has nonzero terms, with coefficients depending on the index — a non-constant law (source: chapter4, §76).
Connection to §68
The recurrent law of §76 matches the §68 law for — the two statements are related by together with the sign flip on (source: chapter4, §76). This is the shadow of a single underlying theorem. Euler does not prove the general law here but says it “can be done so much more easily with the aid of some principles of differential calculus” and, for the moment, treats the agreement with §68 and the many worked examples as evidence enough. See recurrent-series §68.
Identities used later
Euler singles out two forms of the expansion for later use (source: chapter4, §72):
Notable points
- No convergence discussion: Euler works formally, content with the term-to-term consistency.
- The deferral of a rigorous proof to differential calculus (§76) is historically significant — a rigorous binomial theorem for arbitrary exponents was one of the motivating problems for 18th-century analysis.
- The series for specializes to the geometric-series , which is where Euler’s story began.