ζ(3) - Apéry's Constant

ζ(3) = 1 + 1/8 + 1/27 + 1/64 + 1/125 + ...

1.202056903...

The infinite series that equals approximately 1.202056903... We know this number with incredible precision, but we don't know what kind of number it is. Is it algebraic (solution to a polynomial equation) or transcendental (like π and e)? This mystery makes it one of the most intriguing constants in mathematics.

Series Convergence

Animation Speed

Max Terms

Current Term

1.000

Partial Sum

0.202

Error from True Value

Series Terms Visualization

Display Mode

Scale

1/1³ = 1.000000

1/2³ = 0.125000

1/3³ = 0.037037

1/4³ = 0.015625

1/5³ = 0.008000

🔍 The Mystery: What Kind of Number is ζ(3)?

✓ What We Know

It's irrational (proven by Apéry, 1978)
Its value to incredible precision
It appears in quantum field theory
It's related to the distribution of prime numbers

❓ What We Don't Know

Is it algebraic or transcendental?
Does it have a simple closed form?
Its exact relationship to π and e
Its deeper mathematical significance

High-Precision Value

ζ(3) calculated to 50 decimal places:

1.20205690315959428539973816151144999076498629234049

Terms Computed

0.00%

Convergence Rate

Digits of Precision