Topic 10.5 Harmonic Series and p-Series: apply the p-series rule (converges iff p greater than 1) and recognize the harmonic series as the divergent p = 1 case (BC).

What this topic is asking

The College Board (Topic 10.5, BC only) singles out the $p$-series $\sum\frac{1}{n^p}$ and its most famous member, the harmonic series $\sum\frac{1}{n}$. These are the standard benchmark series of the whole unit: you classify them instantly by the value of $p$, and you compare almost everything else to them.

The rule and the borderline

Why the harmonic series diverges despite vanishing terms

The harmonic series is the canonical example that terms going to zero does not guarantee convergence. Its terms $\frac{1}{n}\to 0$, yet the partial sums grow without bound. The integral test proves it: $\int_1^\infty\frac{1}{x}\,dx = \lim_{b\to\infty}\ln b = \infty$, so the series diverges with the integral. A classic grouping argument also shows it: $\frac13 + \frac14 > \frac12$, $\frac15 + \cdots + \frac18 > \frac12$, and so on, adding more than $\frac12$ infinitely often, forcing the sum to infinity. This is the example to cite whenever you need to explain why the $n$-th term test cannot prove convergence, and it anchors your intuition for the $p \le 1$ side of the rule.

A worked classification

Reading p from disguised forms

$p$-series often appear disguised as roots or as constant multiples. Roots convert to fractional powers: $\frac{1}{\sqrt[3]{n^2}} = \frac{1}{n^{2/3}}$, so $p = \frac{2}{3} \le 1$, diverges. Constant multiples do not affect convergence: $\sum\frac{c}{n^p}$ behaves exactly like $\sum\frac{1}{n^p}$ for any nonzero constant $c$, because a finite scalar cannot turn a finite sum infinite or vice versa. So the only thing that matters is the exponent $p$ on $n$ in the denominator. Train yourself to strip away constants and rewrite every root as a power before reading $p$; once $p$ is exposed, the classification is immediate.

Why p-series are the universal benchmark

The reason Topic 10.5 gets its own place is that $p$-series and geometric series are the two families whose convergence you know by inspection, and almost every comparison test (Topic 10.6) compares an unknown series to one of them. When a series has terms that behave like $\frac{1}{n^p}$ for large $n$, you compare it to the corresponding $p$-series; the $p > 1$ rule then settles the question. For example, $\sum\frac{1}{n^2 + 5}$ behaves like $\sum\frac{1}{n^2}$ ($p = 2 > 1$, converges), and $\sum\frac{1}{2n + 3}$ behaves like $\sum\frac{1}{n}$ ($p = 1$, diverges). Memorizing the $p$-series rule cold is therefore the single highest-leverage fact in the convergence-testing toolkit.