Riemann Rearrangement Theorem

The alternating harmonic series converges conditionally to ln(2) ≈ 0.693:

1 − ½ + ⅓ − ¼ + ⅕ − ⅙ + ⋯ = ln(2)

Riemann proved that any conditionally convergent series can be rearranged to converge to any real number—or to diverge.

Target

0.693147

Current Sum

0.000000

+ Terms Used

− Terms Used

Partial Sums — watch the oscillation narrow toward the target

Rearranged Terms

How It Works

The algorithm rearranges terms to approach any target value:

1 If the current sum is below the target, add the next unused positive term (+1/n for odd n).

2 If the current sum is above the target, add the next unused negative term (−1/n for even n).

3 Repeat. Since both sub-series diverge (to +∞ and −∞), we can always overshoot or undershoot as needed, oscillating ever closer to the target.

The key insight: although the full series converges, the positive and negative parts each diverge separately. This gives us infinite "fuel" to push the sum in either direction, allowing convergence to any real number.