Sequences and Their Limits

Calculus 2 · Axiom Academy

LESSON Sequences and Their Limits Understanding how infinite sequences approach finite values through the formal ε-N definition Each term in the sequence has a position n and a value a_n . For example, the sequence gives us: — a separate dot at each whole-number position n . A sequence is a function defined on the positive integers We say a sequence approaches or converges to a limit L if the terms get arbitrarily close to L as n gets large. When n = 10 , a_n = 0.1 (close to 0 ) When n = 100 , a_n = 0.01 (closer to 0 ) When n = 1000 , a_n = 0.001 (even closer to 0 ) But "arbitrarily close" needs to be made precise. That's where the formal ε-N definition comes in. In words: No matter how small an error tolerance you choose, I can find a point in the sequence N after which all terms stay within that tolerance of L . 4. Convergent vs Divergent Sequences A sequence is convergent if it has a finite limit. A sequence is divergent if it doesn't converge to any finite value. You've seen what a sequence is, what it means for one to approach a limit, the precise ε-N test that pins it down, and how convergent and divergent behavior differ. Scroll up to revisit any step.

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