Limit of a Sequence
A sequence is an endless line of numbered values, one for each n. The real question is whether those values settle toward a single destination as n grows larger and larger, and that destination is what we call the limit. If the terms crowd arbitrarily close to one number we say the sequence converges; if they scatter or bounce forever, it diverges. Drag the term-count slider and the dots close in on L = 1; switch sequences and convergence sits beside divergence.
The index n grows without end, and there is no last term to inspect. So the limit of a sequence is not a sum of the first few terms; it asks whether later terms stay near one value. If every term after some index sits inside any band you choose, the sequence converges; if it never sticks to one value, it diverges. Unlike a function limit toward a point a, here you only watch integer indices with n → ∞. Sums, differences, and products of two convergent sequences use each limit, and a quotient is the quotient of the limits when the denominator’s limit is not 0. If two sequences squeeze a third to the same value, the middle one converges there as well: that is the sandwich theorem.
The first sketch is aₙ = 1 + 1/n, with a red dashed line at height L = 1. Gold dots are bright for as many terms as you chose; the rest are dim. Indices run from 1 to 20. Raising the term count lights more dots, and once that count is 5 or more a gold band of width ε = 1/n appears. The same sketch is reused in the ε-N section. There is no separate ε slider; the band width is tied to this n. The comparison sketch switches among three sequences. Mode 0 is 1+1/n, mode 1 is (−1)ⁿ(1+1/n) split above and below, and mode 2 is n/(n+1) sitting toward 1 from below. Each index has one dot in place.
Early terms that are getting larger do not force a sequence to diverge. On the comparison sketch, n/(n+1) keeps growing yet stays below 1. On the other hand, even if sizes look large, a pattern like (−1)ⁿ(1+1/n) that hops between two heights does not gather at one limit. If you use one word for flying off to infinity and for oscillation, you hide whether dots fall one way or bounce between two values. Judging from only the first five terms misses later terms that enter the shrinking band ε = 1/n. For a rational expression you divide by the leading power and watch 1/n; a bounded factor is judged by squeezing from both sides.