Infinite products

An infinite product Product[expr, {k, 1, Infinity}] multiplies infinitely many factors together. Where the summation tutorial turned series into constants, this one turns products into constants — and the answers are, if anything, more surprising: rational products collapse by telescoping, trigonometric products fall out of the Weierstrass factorization of sin and cos, products over the primes reproduce the zeta function, and a handful of exponential products distil the constants e, γ (Euler–Mascheroni), and Glaisher's A out of thin air.

Mathilda evaluates products with a native C cascade under src/product/: a dispatcher (product.c) plus one module per family — rational telescoping, infinite-rational (Gamma), Viète cosine, Cantor double-exponential, Euler prime products, log-sum/Glaisher, geometric, and q-products. Each is also a context-qualified builtin (Product`Viete, Product`EulerPrime, …). Product is HoldAll.

Every transcript was produced by the actual Mathilda binary — and two of the examples below are worked cases where the "textbook" closed form circulating online is subtly wrong; Mathilda (checked with NProduct) gets them right.

From finite to infinite

A finite product with a symbolic bound closes to a factorial or a telescoping form:

In[1]:= Product[k, {k, 1, n}]
Out[1]= Factorial[n]

In[2]:= Product[(k + 1)/k, {k, 1, n}]
Out[2]= 1 + n

In[2] is the archetype of telescoping: (k+1)/k has numerator and denominator that cancel across neighbours, leaving only (n+1)/1. An infinite product is the limit of these partial products — and convergence is stricter than for sums: the factors must approach 1 fast enough that Sum[Log[factor]] converges.

Rational telescoping products

The workhorse family is rational products, where each factor is a ratio of polynomials in k that factor into shifted linear pieces. The Wikipedia telescoping product 1 - 1/k² is the canonical example: writing 1 - 1/k² = ((k-1)(k+1))/k² makes the cancellation visible, and the partial product is (n+1)/(2n), tending to 1/2:

In[1]:= Product[1 - 1/k^2, {k, 2, n}]
Out[1]= (1/2 (1 + n))/n

In[2]:= Product[1 - 1/k^2, {k, 2, Infinity}]
Out[2]= 1/2

The same mechanism handles the quadratic and higher-degree cousins. A quadratic numerator 1 - 2/(k(k+1)) factors as (k-1)(k+2)/(k(k+1)); the difference and sum of cubes factor the cubic case; and a symmetric shift k(k+2)/(k+1)² telescopes directly:

In[1]:= Product[1 - 2/(k*(k + 1)), {k, 2, Infinity}]
Out[1]= 1/3

In[2]:= Product[(k*(k + 2))/(k + 1)^2, {k, 1, Infinity}]
Out[2]= 1/2

For the cubic product, the factorizations Mathilda relies on are visible with Factor:

In[1]:= Factor[k^3 - 1]
Out[1]= (-1 + k) (1 + k + k^2)

In[2]:= Factor[k^3 + 1]
Out[2]= (1 + k) (1 - k + k^2)

The linear parts (k-1)/(k+1) telescope while the quadratic parts (k²+k+1)/(k²-k+1) shift by one index, so the whole product collapses to 2/3:

In[1]:= Product[(k^3 - 1)/(k^3 + 1), {k, 2, Infinity}]
Out[1]= 2/3

When the roots become complex, telescoping is no longer purely rational — the answer picks up a hyperbolic function through the Gamma reflection formula Γ(z)Γ(1-z) = π/sin(πz). Mathilda's Product`RationalInfinite module writes the product as a ratio of Gamma values at the roots and applies the reflection:

In[1]:= Product[(k^2 - 1)/(k^2 + 1), {k, 2, Infinity}]
Out[1]= Pi Csch[Pi]

That is π/sinh(π): the k²-1 part telescopes to a rational, while the k²+1 = (k-i)(k+i) part contributes Γ-values at ±i that reflect into sinh(π).

Trigonometric and hyperbolic factorizations

The deepest source of infinite products is Euler's factorization (1735) of the sine as a product over its roots — later made rigorous by the Weierstrass factorization theorem. Writing the two forms side by side,

sin(π x)/(π x) = ∏ (1 - x²/k²)    and    sinh(π x)/(π x) = ∏ (1 + x²/k²),

each an infinite product over k = 1, 2, 3, ... . Evaluating the sine product at x = 1/2 gives the Wallis product (John Wallis, 1656), the first infinite product ever discovered for π; at x = 1/4 it gives an algebraic multiple of 1/π:

In[1]:= Product[1 - 1/(4*k^2), {k, 1, Infinity}]
Out[1]= 2/Pi

In[2]:= Product[1 - 1/(16*k^2), {k, 1, Infinity}]
Out[2]= (2 Sqrt[2])/Pi

Flipping the sign selects the hyperbolic sine product, and restricting to odd integers selects the hyperbolic cosine:

In[1]:= Product[1 + 1/k^2, {k, 1, Infinity}]
Out[1]= Sinh[Pi]/Pi

In[2]:= Product[1 + 1/(2*k - 1)^2, {k, 1, Infinity}]
Out[2]= Cosh[1/2 Pi]

A different trigonometric mechanism drives Viète's formula (1593) — the oldest infinite product in all of mathematics. It comes from iterating the half-angle identity sin θ = 2 sin(θ/2) cos(θ/2), so that a product of nested cosines telescopes against a single sine. Mathilda's Product`Viete module recognises exactly this cosine double-angle structure:

In[1]:= Product[Cos[Pi/2^(k + 1)], {k, 1, Infinity}]
Out[1]= 2/Pi

A frontier case: the quartic product

Push the telescoping idea to a quartic denominator and you reach the current edge of Mathilda's closed-form engine. The product (k⁴-1)/(k⁴+1) mixes real roots (±1) with the complex roots of k⁴+1, and Mathilda returns it unevaluated:

In[1]:= Product[(k^4 - 1)/(k^4 + 1), {k, 2, Infinity}]
Out[1]= Product[(k^4 - 1)/(k^4 + 1), {k, 2, Infinity}]

It still has a beautiful closed form, obtained by applying the sine and sinh factorizations simultaneously (k⁴-1 uses both sin and sinh, k⁴+1 uses the fourth roots of -1). The result is π sinh(π)/(cosh(√2 π) - cos(√2 π)) — and here is a genuine caution for the reader: several online sources quote this as cosh(π) - cos(π), dropping the √2. NProduct settles the matter to eighteen digits:

In[1]:= NProduct[(k^4 - 1)/(k^4 + 1), {k, 2, Infinity}, WorkingPrecision -> 20]
Out[1]= 0.848054049352900392127

In[2]:= N[Pi*Sinh[Pi]/(Cosh[Sqrt[2]*Pi] - Cos[Sqrt[2]*Pi]), 20]
Out[2]= 0.848054049352900392134

The √2 version matches; the cosh(π) - cos(π) version would give ≈ 2.881, a different number entirely. When a symbolic engine declines to guess, NProduct is how you check which "known" answer is actually correct.

Exponential limits: e, the Euler–Mascheroni constant, and Glaisher

A third family of products has exponential factors and evaluates to the great analytic constants. The cleanest is the exponential of the alternating harmonic series (which sums to log 2), so the product is e^(log(1/2)) = 1/2:

In[1]:= Product[Exp[(-1)^k/k], {k, 1, Infinity}]
Out[1]= 1/2

The next combines a telescoping factor (1 - 1/k) with e^(1/k). The rational part telescopes to 1/N, while Σ 1/k = H_N grows like log N + γ; the two combine so that the log N cancels and the Euler–Mascheroni constant γ survives:

In[1]:= Product[E^(1/k)*(1 - 1/k), {k, 2, Infinity}]
Out[1]= E^(-1 + EulerGamma)

Mind the sign of γ

The partial product is (1/N)·e^(H_N - 1), and since e^(H_N) ≈ e^γ·N, the limit is e^(γ - 1) ≈ 0.6552 — which is what Mathilda returns (E^(-1 + EulerGamma)). It is a common slip to write this as e^(1 - γ); that would be its reciprocal, ≈ 1.526. The telescoping fixes the sign.

The Somos quadratic recurrence constant underlies the rapidly converging 2^(k/2^k), whose exponents Σ k/2^k = 2 sum to give a clean integer:

In[1]:= Product[2^(k/2^k), {k, 1, Infinity}]
Out[1]= 4

Deeper still, k^(1/k²) links directly to the derivative of the zeta function. Taking logarithms gives Σ log(k)/k² = -ζ'(2), so the product is e^(-ζ'(2)). Mathilda's Product`LogSum module reports the equivalent Glaisher–Kinkelin form, using the identity ζ'(2) = (π²/6)(γ + log 2π - 12 log A):

In[1]:= Product[k^(1/k^2), {k, 1, Infinity}]
Out[1]= E^(1/6 Pi^2 (-EulerGamma + 12 Log[Glaisher] - Log[2 Pi]))

A second frontier case: Stirling's product

Closely related to Stirling's approximation of n! is the product (1 + 1/k)^(k + 1/2)/e. Like the quartic product above, Mathilda leaves it symbolically unevaluated — but its true value is a small, memorable rearrangement of Stirling's constant √(2π):

In[1]:= NProduct[(1 + 1/k)^(k + 1/2)/E, {k, 1, Infinity}, WorkingPrecision -> 20]
Out[1]= 1.08443755141922754661

In[2]:= N[E/Sqrt[2*Pi], 20]
Out[2]= 1.08443755141922754661

The value is e/√(2π) ≈ 1.0844. Here too the naive reciprocal √(2π)/e ≈ 0.922 is the version that most often appears in problem sets — and, again, it is wrong. A one-line Stirling estimate of the partial product e^(-N)·∏((k+1)/k)^(k+1/2) pins down the correct orientation.

Euler prime products

The most consequential infinite products in mathematics run not over the integers but over the primes. Euler's 1737 discovery that

ζ(s) = ∏ 1/(1 - p⁻ˢ), the product running over all primes p,

is the analytic gateway to the whole theory of the distribution of primes. Mathilda's Product`EulerPrime module recognises the pattern 1/(1 - Prime[k]^-s) and returns the corresponding zeta value — so the Basel constant reappears, this time as a product over primes:

In[1]:= Product[1/(1 - 1/Prime[k]^2), {k, 1, Infinity}]
Out[1]= 1/6 Pi^2

In[2]:= Product[1/(1 - 1/Prime[k]^4), {k, 1, Infinity}]
Out[2]= 1/90 Pi^4

In[3]:= Product[1/(1 - 1/Prime[k]^6), {k, 1, Infinity}]
Out[3]= 1/945 Pi^6

Twisting the prime product by a Dirichlet character — here the sign (-1)^((p-1)/2), which is +1 for primes p ≡ 1 (mod 4) and -1 for p ≡ 3 (mod 4) — gives the Euler product for the Dirichlet beta function, the prime-side mirror of the Leibniz series for π/4:

In[1]:= Product[1/(1 - (-1)^((Prime[k] - 1)/2)/Prime[k]), {k, 2, Infinity}]
Out[1]= 1/4 Pi

Geometric and Fermat-number products

A final family exploits repeated difference of squares. In the product (1 + x^(2^k)), each factor is exactly what is needed to complete (1 - x)·(1 + x)(1 + x²)(1 + x⁴)... = 1 - x^(2^(N+1)), so the whole product telescopes to 1/(1 - x). Mathilda's Product`Cantor module handles these double-exponential products; with x = 1/3 it gives 3/2:

In[1]:= Product[1 + (1/3)^(2^k), {k, 0, Infinity}]
Out[1]= 3/2

This is the product form of the binary expansion / Fermat-number identity, and it converges doubly exponentially — three factors already give five correct digits.

Where to next

You have now toured Mathilda's full product cascade: rational telescoping, the Gamma-reflection route for complex roots, the sine/sinh/cosine Weierstrass factorizations, Viète's nested cosines, the exponential products for e, γ, and Glaisher's constant, the Euler prime products for ζ and Dirichlet β, and the double-exponential Cantor/Fermat products — plus two honest frontier cases where NProduct is the arbiter of a disputed closed form.

• The companion symbolic summation tutorial covers the parallel machinery for Sum — Gosper telescoping, the zeta family, Euler sums, and the hypergeometric π-machines.
• The numerical calculus tutorial develops NProduct (and NSum) in full, for products with no closed form.
• The special functions tutorial covers Gamma, Zeta, and the constants (EulerGamma, Glaisher, Catalan) these products evaluate to.

As always, ?Product (or ?Product`EulerPrime) at the prompt shows the built-in help without leaving the REPL.