Writing down harmonic numbers
The nth harmonic number is the sum of reciprocals of the first n positive integers. It can be written as a fraction with numerator n! times Hn and denominator n!. This representation is mathematically correct but may not be in lowest terms.
Background
Harmonic numbers (Hₙ) are the sums 1 + 1/2 + 1/3 + … + 1/n. They appear throughout number theory, computer science (analysis of algorithms), and physics, but they have no closed-form formula — you can only approximate them (using the natural log plus the Euler–Mascheroni constant). John D. Cook's blog often explores elegant or surprising mathematical observations with a practical bent. This post asks: if you want to write Hₙ as a fraction, the obvious common denominator is n! (n factorial, the product 1×2×…×n). But that yields absurdly large numerators and denominators even for modest n (e.g., n=10 gives a fraction with denominator 3,628,800). Cook discusses whether there's a smaller denominator — the least common multiple of 1…n — and when you can simplify.