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šŸŽÆ Limits of Sequences Lab

As n grows without bound,
where does a sequence go?

Grow n to a very large value with the slider. Some sequences get closer and closer to a single value (converge), while others keep growing without bound or bounce back and forth without ever settling down (diverge).

A sequence is a list of numbers arranged in order. For example, aā‚™ = 1/n produces the numbers 1, 1/2, 1/3, 1/4... getting smaller and smaller as n gets larger. No matter how large you make n, it never becomes exactly 0, but it can get as close to 0 as you like. When a sequence gets infinitely close to some value L as n grows without bound, we say it converges.

Not every sequence settles on a single value. If it just keeps growing forever, like aā‚™ = n, we call it divergent; if it bounces between two values without ever settling down, like aā‚™ = (āˆ’1)ⁿ, we call it oscillating. Use the slider to grow n up to 1000 or 10000, and directly compare in the table which value each sequence approaches, or whether it never settles down at all.

Convergence: If aā‚™ gets infinitely close to some single value (the limit) L as n grows without bound, this sequence is said to converge to L (written as: limnā†’āˆž aā‚™ = L).
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