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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
A sequence lists terms, its limit is their destination, and monotone bounded lists always arrive.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.