Math Algebra • Sequences and Series
H_n is the partial sum of the harmonic series: H_n = sum_{j=1}^n (1/j). It grows without bound as n increases, but it grows very slowly.
ln(n) + gamma is a standard approximation for H_n, where gamma is the Euler-Mascheroni constant. The approximation improves as n gets larger, and the calculator can show the numeric difference for your chosen n.
The harmonic series sum_{j=1}^infinity (1/j) diverges, meaning its partial sums H_n increase without bound. The divergence is slow, which is why ln(n) comparisons are commonly used.
Generalized harmonic numbers are H_n^(k) = sum_{j=1}^n (1/j^k). For k > 1 the corresponding infinite series converges, while for k = 1 it reduces to the (divergent) harmonic series.
The alternating harmonic series uses terms ((-1)^(j+1)/j) and its partial sums approach a finite limit. This calculator helps you see that convergence by plotting the partial sums and showing the limiting behavior.