Convergence Tests I: The Integral Test, P-Series, and Comparison Tests

The geometric series has a closed-form partial sum, making convergence obvious. Most series do not. The convergence tests in this lesson — the Integral Test, the p-series classification, and the Comparison Tests — extend the reach of the analysis to series whose terms are "almost" functions of $n$ that can be integrated. In quantitative finance, these tests determine whether the moment integrals of heavy-tailed distributions converge: whether $\sum n^{-\alpha}$ (a discrete model for tail probabilities) converges governs the finiteness of portfolio variance under power-law tails. The p-series $\sum 1/n^p$ is the boundary between finite and infinite expected losses.

• apply the Divergence Test as a first screen;
• state and apply the Integral Test, and use it to estimate partial sums;
• classify p-series by their exponent;
• apply the Direct Comparison Test and the Limit Comparison Test.

1. The Integral Test
2. The p-series
3. The Comparison Tests