Dãy Điều hòa và Dãy p

A whole family of test cases

• Two related series come up so often they get names — and a simple convergence rule.
• The harmonic series 调和级数 is $\sum\tfrac1n=1+\tfrac12+\tfrac13+\cdots$.
• The p-series p-级数 generalizes it: $\sum\tfrac1{n^p}$ for a fixed power $p$.
• One clean threshold on $p$ decides convergence for the whole family.

The p-series rule

• $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^{p}}$ converges if $p>1$ and diverges if $p\le 1$.
• Bigger $p$ makes the terms shrink faster — fast enough to converge once $p$ passes $1$.
• This follows directly from the Integral Test on $\int_1^\infty x^{-p}\,dx$.
• Memorize the threshold: the cutoff is exactly $p=1$.

Partial sums of the harmonic series keep climbing; those of Σ1/n² level off

The harmonic series is the borderline

• The harmonic series is the $p=1$ case — and $p=1$ diverges.
• Its terms $\tfrac1n\to0$, yet the sum grows without bound (slowly, but forever).
• It's the classic "terms shrink but the series still diverges" example.
• $p=1$ sits exactly on the wrong side of the cutoff.

Instant convergence checks

• Recognize a p-series and you skip all the work — just compare $p$ with $1$.
• $\sum\tfrac1{n^2}$: $p=2>1$ → converges. $\sum\tfrac1{\sqrt n}=\sum\tfrac1{n^{1/2}}$: $p=\tfrac12\le1$ → diverges.
• These are also the standard comparison series for the comparison tests (next lesson).
• Knowing p-series cold speeds up a huge fraction of series problems.