The harmonic series diverges

Bars representing 1/n shrink toward zero, illustrating why it seems the sum should settle down. The animation then groups terms as 1/3+1/4, 1/5+...+1/8, 1/9+...+1/16, showing each bracket totals at least 1/2. Since infinitely many such groups exist, the running total climbs without bound, proving the harmonic series diverges despite its terms vanishing. Useful for calculus students studying series convergence tests and building intuition beyond the 'terms go to zero' misconception.

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The prompt that made it

Bars of 1/n shrink to zero, yet grouping 1/3 + 1/4, 1/5 + ... + 1/8, 1/9 + ... + 1/16 shows each group is at least 1/2, so the sum grows without bound.

Make your own version

Make the next one in this series

• Compare the harmonic series growth rate to ln(n) using the integral test visually
• Show why 1/n^2 converges while 1/n diverges using stacked rectangle comparisons
• Animate partial sums of the alternating harmonic series converging to ln(2)

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