Sequence limit
A sequence is a list of numbers a_1, a_2, a_3, and so on. Its limit is the value the terms approach as n gets very large.
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Analogy
Walking towards a wall by halving the distance each step: you get closer and closer but never pass it.
Example
n = 1: 3. n = 10: 2.1.n = 100: 2.01.Write as 2 + 1/n.Limit is 2.
In practice
Sequence limits are the starting point for testing whether an infinite sum has a total.
Practice
Flashcards
- What is the limit of the sequence (2n + 1)/n?2.
Questions people ask
What does it mean for a sequence to converge?
A sequence converges if its terms approach one fixed number as n grows. If they grow without bound or keep jumping around without settling, it diverges. The sequence 1, -1, 1, -1 and so on diverges, since it never settles.
How is a sequence different from a series?
A sequence is a list of numbers. A series is the sum of those numbers. The sequence 1, 1/2, 1/4, and so on has limit 0, but the series 1 + 1/2 + 1/4 + and so on adds up to 2. They answer different questions.