Chapter 10: On the Use of the Discovered Factors to Sum Infinite Series

Summary: Euler harvests the Chapter 9 infinite products. The mechanism: an infinite product expansion provides the elementary symmetric polynomials of the function’s reciprocal zeros via its power-series coefficients; Newton’s identities then convert them into power sums. This single technique solves the Basel problem, evaluates [[zeta-at-even-integers|every $\zeta (2k)$]], generates Leibniz’s $\pi /4$ series and a vast family of character-style sums, and produces the partial-fraction expansions of cot, csc, coth, csch.

Sources: chapter10

Last updated: 2026-04-30

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Overview

Chapter 9 built infinite products. Chapter 10 extracts numerical content from them. The recipe is one slide:

1. Take an infinite-product expansion $1+Az+B{z}^2+\cdots =\prod (1+{\alpha }_{k}z)$, where the ${\alpha }_k$ are the reciprocal zeros of the function.
2. The series-side coefficients $A,B,C,\dots$ are the elementary symmetric polynomials of the ${\alpha }_k$.
3. Newton’s identities convert these into the power sums $P=\sum {\alpha }_k$, $Q=\sum {\alpha }_k^2$, $R=\sum {\alpha }_k^3$, $\dots$
4. The power sums are the interesting numbers.

The chapter has four movements:

1. §165–§166 — the engine. Newton’s identities for converting elementary symmetrics to power sums, given as a recurrence: $P=A$, $Q=AP-2B$, $R=AQ-BP+3C$, etc.
2. §167–§170 — even-zeta values. Apply the engine to the sinh product for $\zeta (2)$ (the Basel problem) and through the [[zeta-at-even-integers|table $\zeta (2k)$]] for $k\le 13$. The cosh product gives sums over odd squares; the §170 alternating decomposition connects all variants.
3. §171–§180 — character-style sums. Apply the engine to the §164 arc-form products and obtain the family of series indexed by arithmetic progressions. Special cases: Leibniz’s $\pi /4$, ${\pi }^2/8$, ${\pi }^3/32$; $n=3$ gives $\sqrt{3}$ series; $n=4,8$ gives $\sqrt{2}$ series.
4. §181–§183 — partial fractions. Combine pairs of series and substitute $a=m^2/n^2$ to obtain the partial-fraction expansions $\pi \cot (\pi \sqrt{a})=\cdots$, $\pi /\sin (\pi \sqrt{a})=\cdots$. The hyperbolic version (negative $a$) follows by imaginary substitution.

Structure of the chapter

§165–§166 — Newton’s identities

If $1+Az+B{z}^2+C{z}^3+\cdots =(1+\alpha z)(1+\beta z)(1+\gamma z)\cdots$, then $A,B,C,\dots$ are the elementary symmetric polynomials of $\alpha ,\beta ,\gamma ,\dots$. Defining the power sums $P=\sum {\alpha }_k^1$, $Q=\sum {\alpha }_k^2$, $R=\sum {\alpha }_k^3$, $\dots$, Euler states the recurrence

$P=A,Q=AP-2B,R=AQ-BP+3C,S=AR-BQ+CP-4D,\dots$

(source: chapter10, §166). The truth “is intuitively clear, but a rigorous proof will be given in the differential calculus.” See newtons-identities.

§167 — The Basel problem

Apply the engine to the sinh product $(e^x-e^{-x})/(2x)=\prod (1+x^2/k^2{\pi }^2)$. Substituting $x^2={\pi }^{2}z$ identifies the elementary symmetrics from the power-series coefficients:

$A=\frac{{\pi }^2}{6},B=\frac{{\pi }^4}{120},C=\frac{{\pi }^6}{5040},\dots$

and the roots are ${\alpha }_k=1/k^2$. Newton’s identities deliver

$P={\sum }_{k=1}^{\infty }\frac{1}{k^2}=\frac{{\pi }^2}{6},$

$Q={\sum }_{k=1}^{\infty }\frac{1}{k^4}=\frac{{\pi }^4}{90},R={\sum }_{k=1}^{\infty }\frac{1}{k^6}=\frac{{\pi }^6}{945},S={\sum }_{k=1}^{\infty }\frac{1}{k^8}=\frac{{\pi }^8}{9450},\dots$

See basel-problem and zeta-at-even-integers.

§168 — Tabulation through $\zeta (26)$

Euler tabulates the rational coefficients of $\zeta (2k)/{\pi }^{2k}$ for $k=1,\dots ,13$. The “irregular” sequence $1,1/3,1/3,3/5,5/3,691/105,35/1,\dots$ is, in modern terms, $|B_{2k}|$ (Bernoulli numbers) rescaled. Euler does not have the Bernoulli connection in front of him here, but he notes the sequence’s “extraordinary usefulness.” See zeta-at-even-integers.

§169 — Sums over odd squares

The same mechanism applied to the cosh product $(e^x+e^{-x})/2=\prod (1+4x^2/(2k+1{)}^2{\pi }^2)$ gives

${\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^2}=\frac{{\pi }^2}{8},{\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^4}=\frac{{\pi }^4}{96},{\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^6}=\frac{{\pi }^6}{960},\dots$

See zeta-at-even-integers.

§170 — Even/odd/alternating decomposition

For any $n$ and $M=\zeta (n)=\sum 1/k^n$:

$\frac{M}{2^n}=\sum \frac{1}{(2k{)}^n},M-\frac{M}{2^n}=\frac{2^n-1}{2^n}M=\sum \frac{1}{(2k+1{)}^n},M-\frac{2M}{2^n}=\frac{2^{n-1}-1}{2^{n-1}}M={\sum }_{k\ge 1}\frac{(-1{)}^{k+1}}{k^n}.$

A linear-algebra observation that cleanly converts any closed form for $\zeta (n)$ into closed forms for the even, odd, and alternating restrictions. See odd-and-alternating-zeta-decomposition.

§171–§174 — The arc-form expansion

Substitute $v=\pi x/n$, $g=m\pi /n$ in the §164 product

$\cos (v/2)+\tan (g/2)\sin (v/2)=\prod \left(1\pm \frac{v}{(2j+1)\pi -g}\right)\left(1\mp \frac{v}{(2j+1)\pi +g}\right)$

to obtain

$\left(1+\frac{x}{n-m}\right)\left(1-\frac{x}{n+m}\right)\left(1+\frac{x}{3n-m}\right)\cdots =\cos \frac{\pi x}{2n}+\tan \frac{m\pi }{2n}\sin \frac{\pi x}{2n}.$

The right side, expanded in $x$, gives elementary symmetric coefficients $A,B,C,\dots$ involving $\tan (m\pi /2n)$ and ${\pi }^k/n^k$. Newton’s identities convert them into power sums $P,Q,R,S,T,V$ of the roots $1/(n-m),-1/(n+m),1/(3n-m),\dots$. See circular-arc-series.

§175–§180 — Special values

Euler walks through $m=1,n=2$ (Leibniz’s $\pi /4$ and friends), $m=1,n=3$ ($\sqrt{3}$ series), $m=1,n=4$ and $n=8$ ($\sqrt{2}$ series), and combinations. Each produces a different “character pattern” on the integers. See circular-arc-series for the catalog.

§181–§182 — Pairing two-by-two: $\pi \cot$ and $\pi /\sin$

Adding/subtracting adjacent-term combinations of §172 and §174 collapses to

${\sum }_{k=1}^{\infty }\frac{1}{k^2-a}=\frac{1}{2a}-\frac{\pi }{2\sqrt{a}\tan (\pi \sqrt{a})},{\sum }_{k=1}^{\infty }\frac{(-1{)}^{k+1}}{k^2-a}=\frac{\pi }{2\sqrt{a}\sin (\pi \sqrt{a})}-\frac{1}{2a}.$

These are the partial-fraction expansions of $\pi \cot (\pi \sqrt{a})$ and $\pi /\sin (\pi \sqrt{a})$. See cotangent-partial-fraction.

§183 — Hyperbolic version via Euler’s formula

For $a=-b$, eulers-formula $\cos (yi)=(e^y+e^{-y})/2$, $\sin (yi)=(e^y-e^{-y})/(2i)$ converts the §182 formulas into

${\sum }_{k=1}^{\infty }\frac{1}{k^2+b}=\frac{\pi \sqrt{b}\coth (\pi \sqrt{b})}{2b}-\frac{1}{2b},{\sum }_{k=1}^{\infty }\frac{(-1{)}^{k+1}}{k^2+b}=\frac{1}{2b}-\frac{\pi \sqrt{b}}{2b\sinh (\pi \sqrt{b})}.$

These are the partial fractions of $\coth$ and $\text{csch}$. Euler chooses this route over an independent §162 derivation “since it is a nice illustration of the reduction of sines and cosines of complex arcs to real exponentials” (source: chapter10, §183). See cotangent-partial-fraction.

Notable points

• Newton’s identities are the only computational engine. The whole chapter is one technique applied to many products. This is unusual for Euler, who normally varies methods; here the technique’s range is the point.
• Even-zeta values come for free from Chapter 9. The Basel problem — open for a century, the showpiece of the Introductio — falls out of $A={\pi }^2/6$ in the sinh product. No further work.
• Odd-zeta values are absent. The chapter never produces $\zeta (2k+1)=\sum 1/n^{2k+1}$ in closed form. This is not a gap in Euler’s exposition; it is a structural limit of the method, and the limit persists in 2026.
• Character sums emerge from arc parameters. The §171–§180 family — $1-1/3+1/5-1/7+\cdots$, $1-1/2+1/4-1/5+\cdots$, and many more — are the values of [[circular-arc-series|Dirichlet $L$-functions]] at integer arguments, computed before Dirichlet introduced the concept. Euler’s parametrization $(m,n)$ corresponds to a real quadratic character mod $4n$.
• Partial fractions arise from product manipulation. The §181–§183 derivation of $\pi \cot$ and $\pi /\sin$ partial fractions is the first appearance of what would become Mittag-Leffler theory. Euler reaches the formulas without complex analysis as a discipline; they are correct and survived unaltered into modern texts.
• Euler uses real and imaginary forms interchangeably. §183 transitions between trigonometric and hyperbolic series by setting $a=-b$, with eulers-formula as the dictionary. This freedom — rotating across the imaginary axis to swap circular for hyperbolic — is one of the [[chapter-7-on-exponentials-and-logarithms-expressed-through-series|Introductio’s]] characteristic moves.

Why this chapter matters

Chapter 10 closes a chord struck across the entire first volume:

• Chapter 2: every polynomial factors over $\mathbb{R}$.
• Chapter 7 / Chapter 8: the elementary transcendentals $e^z$, $\sin z$, $\cos z$ are limits of polynomials.
• Chapter 9: factoring those polynomials through the limit produces infinite products.
• Chapter 10: comparing the products to the power series produces every numerical result of importance — ${\pi }^2/6$, ${\pi }^4/90$, $\pi /4$, $\pi \cot (\pi z)$, $\pi /\sin (\pi z)$.

Each step uses only the previous one. The whole edifice rests on real algebra, infinitesimals, and the imaginary substitution — Euler’s three persistent tools.

After Chapter 10, the Introductio’s analytic-function project is complete. Chapters 11–18 specialize to recurring topics (continued fractions, partition identities, Diophantine equations) that complement but do not extend the synthesis.

Related pages

• newtons-identities
• basel-problem
• zeta-at-even-integers
• odd-and-alternating-zeta-decomposition
• circular-arc-series
• cotangent-partial-fraction
• exponential-infinite-product
• sine-infinite-product
• cosine-infinite-product
• eulers-formula
• arctangent-series
• chapter-9-on-trinomial-factors
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle