Wallis Product

Summary: §185: $\frac{\pi }{2}=\frac{2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6\cdot 8\cdot 8}{1\cdot 3\cdot 3\cdot 5\cdot 5\cdot 7\cdot 7\cdot 9}\cdots$. Euler derives Wallis’s 1656 product as the quotient of two §184 product expressions for $\cos (m\pi /2n)$, embedding it in a parametric family that includes analogous products for $\sqrt{2}$ and other algebraic numbers.

Sources: chapter11

Last updated: 2026-05-11

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Statement

$\boxed{\frac{\pi }{2}=\prod _{k=1}^{\infty }\frac{(2k)(2k)}{(2k-1)(2k+1)}=\frac{2\cdot 2}{1\cdot 3}\cdot \frac{4\cdot 4}{3\cdot 5}\cdot \frac{6\cdot 6}{5\cdot 7}\cdot \frac{8\cdot 8}{7\cdot 9}\cdots }$

Equivalently, taking adjacent factors together:

$\frac{\pi }{2}=\frac{2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6\cdot 8\cdot 8\cdot 10\cdot 10\cdot 12\cdot 12}{1\cdot 3\cdot 3\cdot 5\cdot 5\cdot 7\cdot 7\cdot 9\cdot 9\cdot 11\cdot 11\cdot 13}\cdots$

(source: chapter11, §185). Euler attributes the formula directly: “this is the expression for $\pi$ which Wallis found in his Arithmetic of the Infinite” (Wallis, Arithmetica Infinitorum, 1656).

Derivation as a quotient

The §184 cosine identities give two product expressions for $\cos (m\pi /2n)$:

$\cos \frac{m\pi }{2n}=\frac{n-m}{n}\cdot \frac{n+m}{n}\cdot \frac{3n-m}{3n}\cdot \frac{3n+m}{3n}\cdot \frac{5n-m}{5n}\cdot \frac{5n+m}{5n}\cdots$

$\cos \frac{m\pi }{2n}=\frac{(n-m)\pi }{2n}\cdot \frac{n+m}{2n}\cdot \frac{3n-m}{2n}\cdot \frac{3n+m}{4n}\cdot \frac{5n-m}{4n}\cdot \frac{5n+m}{6n}\cdots$

Dividing the first by the second cancels every $(n-m),(n+m),(3n-m),\dots$ numerator and produces

$1=\frac{\pi }{2}\cdot \frac{1}{2}\cdot \frac{3}{2}\cdot \frac{3}{4}\cdot \frac{5}{4}\cdot \frac{5}{6}\cdot \frac{7}{6}\cdot \frac{7}{8}\cdot \frac{9}{8}\cdots$

(source: chapter11, §185), independent of $m$ and $n$. Solving for $\pi /2$ gives the Wallis product. The $m,n$ drop out: the identity is a structural fact about the §184 redundancy, not about any particular angle.

Variants for other algebraic numbers

The general form (source: chapter11, §185) is

$\frac{\pi }{2}=\frac{n}{m}\sin \frac{m\pi }{2n}\cdot \frac{2n}{2n-m}\cdot \frac{2n}{2n+m}\cdot \frac{4n}{4n-m}\cdot \frac{4n}{4n+m}\cdot \frac{6n}{6n-m}\cdots$

Setting $m/n=1$ recovers Wallis. Other rationals give:

$m/n=1/2$ ($\sin (\pi /4)=1/\sqrt{2}$):

$\frac{\pi }{2}=\frac{\sqrt{2}}{1}\cdot \frac{4}{3}\cdot \frac{4}{5}\cdot \frac{8}{7}\cdot \frac{8}{9}\cdot \frac{12}{11}\cdot \frac{12}{13}\cdot \frac{16}{15}\cdot \frac{16}{17}\cdots$

$m/n=1/3$ ($\sin (\pi /6)=1/2$):

$\frac{\pi }{2}=\frac{3}{2}\cdot \frac{6}{5}\cdot \frac{6}{7}\cdot \frac{12}{11}\cdot \frac{12}{13}\cdot \frac{18}{17}\cdot \frac{18}{19}\cdot \frac{24}{23}\cdots$

Dividing the first by the second (both equal $\pi /2$) eliminates $\pi$ entirely:

$\sqrt{2}=\frac{2\cdot 6\cdot 6\cdot 10\cdot 10\cdot 14\cdot 14\cdot 18\cdot 18}{1\cdot 3\cdot 5\cdot 7\cdot 9\cdot 11\cdot 13\cdot 15\cdot 17\cdot 19}\cdots$

(source: chapter11, §185). A “Wallis-style” product for $\sqrt{2}$.

Convergence is slow

The $k$-th Wallis factor is $1+1/(4k^2-1)=1+O(1/k^2)$, so the partial product converges to $\pi /2$ at rate $1/N$. Concretely:

Factors	Approximation to $\pi /2$
10	1.55340 (3 correct digits)
100	1.56688 (3 correct digits)
1000	1.56999 (4 correct digits)
10000	1.57072 (5 correct digits)

Euler is explicit: “too many terms are required to obtain an accurate value of $\pi$ even to only ten decimal places” (source: chapter11, §188). The Wallis product is structurally important but computationally inferior to Machin’s formula (chapter 8) or the [[arctangent-series|$\arctan$ series]].

What it is good for

The Wallis product (and its variants) is the form in which the §158 products land at rational angles, and that form makes its logarithm tractable. Re-pairing factors gives

$\frac{\pi }{2}=2{\prod }_{k=1}^{\infty }\left(1-\frac{1}{(2k+1{)}^2}\right)=2\cdot \frac{8}{9}\cdot \frac{24}{25}\cdot \frac{48}{49}\cdot \frac{80}{81}\cdots$

— each factor is $(1-$ small$)$, so $\log \pi =\log 4+\sum \log (1-1/(2k+1{)}^2)$ has every term tame. The double-sum transposition that follows (§188–§190) extracts $\log \pi$ to twenty digits. See log-pi-via-products.

Modern footnote

Wallis’s original proof (1656) was a heroic interpolation of integrals of $(1-x^2{)}^n$ at half-integer $n$, giving him ${\int }_0^1\sqrt{1-x^2}dx=\pi /4$. Euler’s derivation here — quotient of two ostensibly different products for the same trig value — is structurally cleaner and embeds Wallis in a parametric family. The full family is the start of the theory of infinite products with positive factors and is essentially a consequence of the $\Gamma$-function duplication formula, though Euler does not yet have $\Gamma$ in the Introductio.

Related pages

• linear-factors-of-sine-cosine
• sine-infinite-product
• cosine-infinite-product
• log-pi-via-products
• trig-infinite-products
• pi
• machin-like-formula
• arctangent-series
• chapter-11-on-other-infinite-expressions-for-arcs-and-sines
• chapter-15-on-series-which-arise-from-products
• prime-sign-series-for-pi
• euler-product-formula