Consider the sequence aₙ = 1/n. a₁=1, a₅=0.2, a₁₀=0.1, a₁₀₀=0.01, a₁₀₀₀=0.001... as n gets larger, the value gets smaller and smaller, getting closer to 0. Of course, it never actually becomes 0 (no matter how large a number you divide 1 by, it's never exactly 0). But you can grow n as large as you like, and aₙ can get as close to 0 as you like. In this case, we say "the sequence aₙ converges to 0," and write limn→∞ aₙ = 0.
Not every sequence settles down onto a single value like this. Look at aₙ = n, and you get 1, 2, 3, 4... just growing forever. Since it doesn't approach any value and just keeps growing without bound, we say this sequence diverges (more precisely, "diverges to positive infinity"). There's another type of divergence too. Look at aₙ = (−1)ⁿ, and you get −1, 1, −1, 1... bouncing between two values forever, never settling on a single one. A case like this, where it neither grows nor converges, is called oscillating.
There's a trick for judging whether a sequence written as a fraction converges. An expression like aₙ = (2n+1)/(n+3) can be rewritten by dividing both the numerator and denominator by n, giving the form (2+1/n)/(1+3/n). As n grows without bound, both 1/n and 3/n get closer to 0, so in the end the fraction gets closer to 2/1 = 2. Just by comparing the coefficients of the highest-degree terms in the numerator and denominator (here, 2 and 1), you can quickly predict the limit.
The limit of a sequence is, in fact, essentially the same concept as the limit of a function, which you learn later. There's exactly one difference: a sequence only takes natural-number values of n (1, 2, 3, ...). The limit of a function deals with x smoothly moving through all real numbers as it approaches some value, while the limit of a sequence can be thought of as the special case that only picks out the integer points among those. That's why the intuition for convergence and divergence you build up with sequence limits carries straight over when you learn function limits.
On our activity page, you can pick one sequence out of several, grow n larger and larger with a slider, and directly compare in a table whether the value approaches a certain number (convergence) or keeps growing or bouncing around (divergence). Try growing n all the way to 1000 and see whether the value really settles down.