Sequences and Limits
Convergence of infinite lists of numbers.
Key idea
Convergence of infinite lists of numbers.
In depth
A sequence is an ordered list aₙ. It converges if aₙ approaches a finite number L as n → ∞. Bounded monotonic sequences always converge.
Worked example
aₙ = (2n + 1)/n → 2. And aₙ = (−1)ⁿ does not converge.
Real-world uses
- Understanding whether a repeated calculation settles down, such as a converging iterative estimate
Common mistake
Concluding convergence because the terms look like they are getting close for the first few values.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Sequences and Limits?
Convergence of infinite lists of numbers. It is a topic at the advanced level in Calculus.
Where is Sequences and Limits used in real life?
Real-world examples: Understanding whether a repeated calculation settles down, such as a converging iterative estimate.
What should I learn before Sequences and Limits?
Make sure you are comfortable with Simpson’s Rule, the previous topic in Calculus.
What is a common mistake with Sequences and Limits?
Concluding convergence because the terms look like they are getting close for the first few values.
How can I practise Sequences and Limits?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Sequences and Limits?
Move on to Infinite Series next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.