Harmonic series diverges

Statement

The partial sums $H_N = \sum_{n=1}^{N} \frac{1}{n}$ of the harmonic series tend to $+\infty$ as $N \to \infty$. In particular, the harmonic series is a divergent series.

Remarks

The classical proof groups the terms in blocks from $2^{k-1} + 1$ to $2^k$. Each block has $2^{k-1}$ terms, each at least $1/2^k$, so each block sums to at least $1/2$. Thus $H_{2^k} \ge 1 + k/2$. A second proof uses the integral test: $H_N \ge \int_1^{N+1} dx/x = \ln(N + 1)$. Mathlib states it as the limit of $\sum_{n < N} 1/(n+1)$, in Real.tendsto_sum_range_one_div_nat_succ_atTop.

History

Nicole Oresme proved the divergence in the 14th century, with the grouping argument. Pietro Mengoli and Jacob Bernoulli gave other proofs in the 17th century.

Explanations

L1 human

Group the terms: $1/3 + 1/4 \ge 1/2$, then $1/5 + \cdots + 1/8 \ge 1/2$, then $1/9 + \cdots + 1/16 \ge 1/2$, and so on. There are infinitely many such groups, so the sum exceeds each bound.