--- title: "Sequences and Their Limits" description: "Treat a sequence as a function on the positive integers and ask what its terms approach. Monotone bounded sequences converge, which is the first place completeness is doing real work." canonical: https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc source: https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc.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`. # Sequences and Their Limits Treat a sequence as a function on the positive integers and ask what its terms approach. Monotone bounded sequences converge, which is the first place completeness is doing real work. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc ## Ready when they can - Decide convergence for a sequence given by a formula in n - Show a recursively defined sequence is increasing and bounded above - Distinguish a sequence from the series formed by summing it ## Lesson: Where sequences settle down A sequence is an ordered list of numbers indexed by n, and its limit is where the terms head as n grows. For n over n plus 1, divide top and bottom by n to get 1 over 1 plus 1 over n, which tends to 1. Leading coefficients rule the same way: 2n over n plus 1 tends to 2, and 5n minus 1 over 2n plus 3 tends to 2.5. Signs need not spoil the trip: minus 1 to the n over n is squeezed between minus 1 over n and 1 over n, so it tends to 0. **Example.** Some sequences come from a rule, not a formula. Start with first term 1, and build each next term as the current term plus 4, over 3. Check the base: 1 sits below 2. Then assume some term sits at most 2, so the next is at most 2 plus 4 over 3, which is 2, and induction keeps every term below 2. The terms also climb without ever crossing that ceiling, so they must settle somewhere. A deeper rule can lead to L squared equals 2 plus L, which factors into L minus 2 times L plus 1, and the positive root is 2. A series is a different animal: it adds the terms instead of listing them. Start with a sequence, form running totals called partial sums, and the series is the limit of those totals. So a list settling down and its running total settling down are two separate questions. Mixing them up is the most common slip in this chapter. **Tip.** Attack a formula by dividing through by the highest power of n. For an alternating term with shrinking size, trap it with the squeeze. For a rule-defined sequence, prove it climbs and has a ceiling, then solve the fixed-point equation for the limit. **Recap.** A sequence lists terms, its limit is their destination, and monotone bounded lists always arrive. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Geometric Sequences and Common Ratios](https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY) ## Opens up - [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt) - [The Completeness Axiom: Suprema and Infima](https://lightmysky.com/learn/mathematics/the-completeness-axiom-suprema-and-infima-mt_xjI-pIfh95)