Unit 15: Sequences and Their Limits
A list of numbers marching toward infinity, and the handful of ways to prove whether it settles or runs away.
5 topics · 20 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Defining a Sequence and Its Limit
- How it worksHow does the formal limit definition pin down what it means for a sequence to get arbitrarily close to a number?
- Worked exampleFind the limit of the sequence an = (3n + 1)/(2n - 5) as n approaches infinity.
- Worked exampleFind the limit of the sequence an = n/en as n approaches infinity using L'Hopital's rule on the matching function.
- Common mistakeWhy doesn't a sequence need to actually reach its limit for the limit to exist?
Your turn1 of 2
Find the limit of the sequence an = (4n + 4)/(6n - 2) as n approaches infinity.Sequences Defined Recursively
- How it worksHow does a recursive rule like a_(n+1) = f(an) generate every term of a sequence from just the one before it?
- Worked exampleFind the first five terms of the sequence defined by a1 = 2, a_(n+1) = √(2 + an).
- Common mistakeWhy doesn't a recursively defined sequence need infinitely many starting values instead of just one?
Your turn1 of 2
For the sequence a1 = 3, a_(n+1) = √(4 + an), find a3.Monotonic and Bounded Sequences
- How it worksHow do monotonicity and boundedness together guarantee convergence even without finding the limit's exact value?
- Worked exampleShow me that the sequence an = n/(n + 1) is increasing and bounded above by 1.
- Going furtherWhat happens to a monotonic sequence that is bounded below instead of above?
- Common mistakeWhy doesn't a bounded sequence have to converge if it isn't also monotonic?
Your turn1 of 2
Find the 8th term of the sequence an = 4n/(n + 4).The Squeeze Theorem for Sequences
- How it worksHow do two sequences converging to the same limit force any sequence squeezed between them there too?
- Worked exampleFind the limit of an = sin(n)/n as n approaches infinity using the squeeze theorem.
- Common mistakeWhy doesn't the squeeze theorem require the trapped sequence to be monotonic at all?
- Going furtherWhat happens when the two bounding sequences converge to different limits instead of the same one?
Your turn1 of 2
Using the squeeze theorem, what is the limit of an = sin(n)/n as n approaches infinity?Sequences Versus Series
- How it worksHow does building the sequence of partial sums turn a series into just another sequence with its own limit?
- Worked exampleFind the first four partial sums of the series formed by the sequence an = 1/2n.
- Worked exampleFind the tenth partial sum of the series formed by the sequence an = 1/n.
- Common mistakeWhy isn't the limit of a series' terms the same thing as the sum of the series?
- Ties togetherHow does the language of sequences and their limits carry over into the definition of a convergent series?
Your turn1 of 2
Find the fourth partial sum of the series formed by the sequence an = 1/5n.