Concept preview

What you will be able to do

1. 01Recognise a p-series $\sum_{n=1}^{\infty}\dfrac{1}{n^p}$ from its algebraic form and read off the exponent $p$, even when the term is disguised as a root such as $\dfrac{1}{\sqrt{n}}$ or $\dfrac{1}{\sqrt[3]{n^2}}$.
2. 02State and apply the p-series test: $\sum \dfrac{1}{n^p}$ converges if and only if $p>1$, and diverges if and only if $p\le 1$.
3. 03Explain why the harmonic series ($p=1$) diverges even though its terms shrink to zero, and use that fact to avoid the most common convergence error on the AP exam.

Fix a positive number $p$ and add up the reciprocals of the $p$-th powers of the whole numbers: $\sum_{n=1}^{\infty}\dfrac{1}{n^p}=\dfrac{1}{1^p}+\dfrac{1}{2^p}+\dfrac{1}{3^p}+\dfrac{1}{4^p}+\cdots.$ This is the p-series. The number $p$ controls how quickly the terms shrink. When $p$ is large the terms fall off steeply and the sum settles to a finite value; when $p$ is small the terms decay lazily and the running total escapes to infinity. The entire behaviour is captured by one threshold.