Chapter 13: On Recurrent Series

Summary: Euler completes the theory of recurrent series introduced in Chapter 4. Three movements: (i) closed-form general term of any recurrent series, obtained by real partial fractions of the generating rational function plus explicit formulas for each fraction type; (ii) the inverse problem — extracting the generating rational function from the recurrence law — and De Moivre’s “scale of the relation”; (iii) the sum of a recurrent series, finite or infinite, expressed in closed form.

Sources: chapter13, chapter17

Last updated: 2026-05-11

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Overview

Chapter 4 introduced recurrent series — power series whose coefficients obey a fixed linear recurrence — and showed that every proper rational function expands into one. The coefficient law was read directly off the denominator. But the recurrence by itself does not give the $n$-th coefficient as a closed-form function of $n$. Chapter 13 supplies that closed form, by combining three pieces:

• Chapter 2’s decomposition into simple linear-factor fractions,
• Chapter 12’s extension to real quadratic factors,
• explicit power-series expansions of each fraction type, then summed.

The chapter also reverses the question (given the series, recover the rational function — the scale of the relation) and computes both finite and infinite sums.

Structure of the chapter

§211–§214 — Decompose, expand, sum

Any proper rational function decomposes by Chapters 2 and 12 into a sum of simple fractions. Each simple fraction expands into its own recurrent series. The general term of the original series is the sum of the general terms of the partial-fraction series (source: chapter13, §212–§213). Equality of two power series is justified by setting $z=0$, subtracting the constant, dividing by $z$, repeating (source: chapter13, §214).

§215–§216 — General term for $A/(1-pz{)}^k$

The fundamental linear-factor brick is

$\frac{A}{(1-pz{)}^k}=A+kApz+\frac{k(k+1)}{2!}A{p}^2{z}^2+\cdots ,$

with general term

$\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)A{p}^n{z}^n=\frac{(n+1)(n+2)\cdots (n+k-1)}{(k-1)!}A{p}^n{z}^n.$

(source: chapter13, §215). Once this is in hand, the partial-fraction tower for any repeated linear factor gives the general term term-by-term (§216). See general-term-of-recurrent-series.

Examples I–V (§216, p. 183–186)

Five worked examples illustrate the method for distinct or repeated real linear factors:

• I. $\frac{1-z}{1-z-2z^2}$ has partial fractions $\frac{2/3}{1+z}+\frac{1/3}{1-2z}$; general term $\frac{2^n\pm 2}{3}{z}^n$ (sign by parity of $n$).
• II. $\frac{1-z}{1-5z+6z^2}=\frac{-1}{1-2z}+\frac{2}{1-3z}$; general term $(2\cdot 3^n-2^n)z^n$.
• III. $\frac{1+2z}{1-z-z^2}$ has roots $(1\pm \sqrt{5})/2$; general term is the sum of $((1+\sqrt{5})/2{)}^{n+1}+((1-\sqrt{5})/2{)}^{n+1}$ — the Lucas-numbers Binet formula.
• IV. Symbolic: $\frac{a+bz}{1-\alpha z-\beta z^2}$ — the closed form for any two-term linear recurrence (Binet-type formula).
• V. $\frac{1}{1-z-z^2+z^3}=\frac{1}{(1-z{)}^2(1+z)}$, with repeated factor; partial fractions $\frac{1/2}{(1-z{)}^2}+\frac{1/4}{1-z}+\frac{1/4}{1+z}$ give general term $\frac{2n+3\pm 1}{4}{z}^n$.

§217–§222 — General term for the quadratic-factor brick

When the denominator has the trinomial factor $1-2pz\cos \varphi +p^2{z}^2$, complex partial fractions expose a closed form involving sines and cosines of multiples of $\varphi$. The base case $k=1$ comes from the cos recurrent progression: the series for $A/(1-2pz\cos \varphi +p^2{z}^2)$ has general term $\frac{A\sin (n+1)\varphi }{\sin \varphi }{p}^n{z}^n$ (source: chapter13, §218). For $A+Bpz$ in the numerator, this becomes $\frac{A\sin (n+1)\varphi +B\sin n\varphi }{\sin \varphi }{p}^n{z}^n$.

For $k=2$ (§220) and $k=3$ (§221) Euler derives explicit, increasingly complicated formulas; the §219 derivation passes through a complex factorization $(1-(\cos \varphi +i\sin \varphi )pz)(1-(\cos \varphi -i\sin \varphi )pz)$ and converts back to real form via the identities $16{\sin }^5\varphi =10\sin \varphi -5\sin 3\varphi +\sin 5\varphi$ and similar (§222) — the same odd-power table that Chapter 14 derives systematically in §262. See general-term-of-recurrent-series.

§223 — Two big mixed examples

The two examples sweep up everything in the chapter:

• $\frac{1}{(1-z)(1-z^2)(1-z^3)}=\frac{1}{(1-z{)}^3(1+z)(1+z+z^2)}$ — repeated linear, simple linear, and quadratic factor (with $\varphi =\pi /3$). The general term takes a different closed form on each residue class $n(\mathrm{mod}6)$ (source: chapter13, §223 Example I).
• $\frac{1+z+z^2}{1-z-z^4+z^5}=\frac{1+z+z^2}{(1-z{)}^2(1+z)(1+z^2)}$ — produces a four-case formula, one per residue class mod $4$.

§224–§230 — The inverse problem: scale of the relation

Reading the rational function back off the recurrence law: given $D=\alpha C+\beta B+\gamma A$ etc., the denominator is $1-\alpha z-\beta z^2-\gamma z^3$. The list $\alpha ,\beta ,\gamma ,\dots$ is De Moivre’s [[scale-of-the-relation|scale of the relation]] (source: chapter13, §224).

For a two-member scale (each term determined by the two preceding), §226–§229 give the Binet-type closed form $A_n=(U{p}^n+V{q}^n)$ where $p,q$ are the roots of the denominator. A striking identity follows: $UV=\frac{B^2-\alpha AB+\beta A^2}{4\beta -{\alpha }^2}$ (source: chapter13, §227). Hence each term can be obtained from a single predecessor by

$Q=\frac{1}{2}\alpha P+\sqrt{((1/4){\alpha }^2-\beta )P^2+(B^2-\alpha AB+\beta A^2){\beta }^n}$

— an apparent irrationality that is always rational. Worked Lucas example (§229 Example): $Q=\frac{1}{2}(P+\sqrt{5P^2+20})$, sign by parity. Section §230 sketches the analogous cubic relation for a three-member scale.

§231–§233 — Sum of a recurrent series

The sum of the infinite series equals the generating rational function (source: chapter13, §231). The sum of the first $n+1$ terms is the rational function minus the tail, which is itself a rational function with shifted numerator (source: chapter13, §232). For a two-member scale this collapses to a clean closed form

${\sum }_{k=0}^n{A}_k{z}^k=\frac{A+(B-\alpha A)z-Q{z}^{n+1}-R{z}^{n+2}}{1-\alpha z-\beta z^2}.$

Lucas example at $z=1$: $1+3+4+7+11+\cdots +P=P+Q-3=\frac{3P-6+\sqrt{5P^2+20}}{2}$ — the partial sum is determined by the last term alone (source: chapter13, §233 Example). See sum-of-recurrent-series.

Notable points

• The chapter is the closed-form complement to Chapter 4. Chapter 4 gave the recurrence; Chapter 13 gives the closed form. The bridge between them is the partial-fraction decomposition of the generating rational function.
• Real-partial-fraction machinery powers the trigonometric formulas. §218’s clean $\sin (n+1)\varphi /\sin \varphi$ is the "$k=1$" case of the iterative tower in §219–§222, which extends to repeated quadratic factors. The Chapter 12 tower is exactly what makes the extension possible.
• De Moivre’s name appears twice. Once for naming the recurrent series themselves (§211, restating §62), once for naming the scale of the relation (§224). De Moivre’s Miscellanea Analytica (1730) is the proximate source.
• The $UV$ identity is the discriminant-like invariant. §227’s $UV=(B^2-\alpha AB+\beta A^2)/(4\beta -{\alpha }^2)$ is the “irrational core” of the closed-form: it isolates exactly the quantity that appears under the square root in the term-from-predecessor formula. Modulo the substitution $\beta \mapsto -\beta$ this is a standard Fibonacci-like invariant.
• Why apparent irrationality stays rational. The series coefficients are rational by construction; the square root in §227 must therefore always evaluate to a rational. Euler does not prove this — he just notes it (source: chapter13, §227).

Why this chapter matters

Closed-form general terms make recurrent series a tractable computational tool: any coefficient can be evaluated directly without iterating the recurrence. The trigonometric formulas of §217–§222 anticipate the discrete Fourier-type structure of solutions to linear recurrences with complex roots (modern: oscillatory solutions $r^n\cos n\varphi$, $r^n\sin n\varphi$). The “scale of the relation” framework is essentially the modern theory of constant-coefficient linear difference equations, written half a century before Lagrange formalized the analogous theory for differential equations.

Combined with Chapter 12’s real partial fractions and Chapter 4’s expansion machinery, the Introductio now has a complete, real, closed-form theory of rational generating functions — the prerequisite for both the integration of rational functions and the formal manipulation of generating series in combinatorics.

Inverse application in Chapter 17

Chapter 17 runs the chapter-13 closed-form machinery in reverse. There, the equation’s coefficients are known but the roots are not; the dominant term $U{p}^n$ of the §215 expansion controls the ratio $Q/P\to p$ of consecutive recurrent-series coefficients, giving Daniel Bernoulli’s method for the largest root. The §217–§222 trigonometric formulas play the same role for a dominant trinomial factor — Euler eliminates $A,B,$ and $n$ from the §218 closed form to extract the modulus $p$ and argument $\varphi$ of a dominant complex pair from four consecutive coefficients (trinomial-factor-from-recurrent-series).

Related pages

• recurrent-series
• general-term-of-recurrent-series
• scale-of-the-relation
• closed-form-two-term-recurrence
• sum-of-recurrent-series
• partial-fraction-decomposition
• real-partial-fraction-decomposition
• trinomial-factor
• trigonometric-recurrent-progression
• de-moivre-formula
• powers-of-sine-and-cosine
• chapter-4-on-the-development-of-functions-in-infinite-series
• chapter-12-on-the-development-of-real-rational-functions
• chapter-14-on-the-multiplication-and-division-of-angles
• bernoullis-method-for-roots
• trinomial-factor-from-recurrent-series
• chapter-17-using-recurrent-series-to-find-roots-of-equations