--- title: "Convergence Tests for Series of Positive Terms" description: "Decide convergence with the integral, comparison, limit comparison, ratio and root tests, and choose the test that matches the shape of the terms." canonical: https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8 source: https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Convergence Tests for Series of Positive Terms Decide convergence with the integral, comparison, limit comparison, ratio and root tests, and choose the test that matches the shape of the terms. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8 ## Ready when they can - Match a p-series or a factorial to the test that settles it fastest - Use limit comparison against a known series and justify the comparison - Say what the ratio test leaves undecided when its limit equals 1 ## Lesson: Matching each series to its test The p-series rule is your fastest friend: the sum of 1 over n to the p converges exactly when p exceeds 1. So 1 over n cubed converges at a glance, with p equals 3. Roots hide powers too: n times square root of n is n to the 1.5, so that series converges with p equals 1.5. **Example.** Lena faces 1 over n squared plus 5 and stands it next to 1 over n squared. Each of her terms sits below the benchmark, and the benchmark converges with p equals 2, so hers converges by direct comparison. For 1 over n squared plus 1, the ratio against 1 over n squared simplifies to n squared over n squared plus 1, which tends to 1. A finite positive limit means both series share the same fate, so limit comparison settles it too. Factorials and exponentials belong to the ratio test, which watches consecutive terms. A limit below 1 gives convergence, above 1 gives divergence, and exactly 1 says nothing at all. The harmonic series returns exactly 1, which is silent, and only another test shows it diverges. In fact every p-series returns 1, so reaching for ratio on plain powers wastes time. **Tip.** Match the shape to the test: plain powers use the p-series rule, cluttered fractions use comparison, and factorials with exponentials use ratio. Keep only the dominant part of the term when you pick a benchmark. If ratio hands you 1, drop it and try something else. **Recap.** Powers use p, clutter uses comparison, factorials use ratio, and a ratio of 1 decides nothing. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt) - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z) ## Opens up - [Alternating Series and Absolute Convergence](https://lightmysky.com/learn/mathematics/alternating-series-and-absolute-convergence-mt_sk6F4n1USH)