Why does the harmonic series diverge?
Its terms approach zero, but they do so too slowly for the accumulated sum to settle.
LaTeX article Updated July 11, 2026
Why the nth-term test is one-way
A convergent series must have terms approaching zero, so a nonzero term limit proves divergence. But zero is only necessary, not sufficient. The harmonic series is the canonical counterexample.
Series convergence depends on accumulated tail mass, not on individual terms alone.
The grouping proof
Group terms after the first into blocks whose lengths double: two terms, four terms, eight terms, and so on. In each block, every term is at least as large as the block’s last term.
Each block therefore contributes at least 1/2. Infinitely many half-units force the partial sums beyond every finite bound.
Comparison perspective
The integral of 1/x from 1 to infinity also diverges, giving an independent integral-test proof. Both arguments show the reciprocal decay is exactly too slow.
Worked example
Common mistakes
- Claiming the series converges because 1/n approaches zero.
- Confusing a sequence of terms with the sequence of partial sums.
- Treating slow divergence as numerical convergence after a finite computation.
Keep these ideas
- Term limits can prove divergence but not convergence.
- Grouping exposes persistent tail mass.
- Partial sums, not terms, determine series convergence.