Sequences & series
From Sequences & series
A sequence is an ordered list of numbers written as a_1, a_2, a_3, ..., where a_n describes the term in position n. The sequence converges to a limit L when its terms can be made arbitrarily close to L by taking n sufficiently large; otherwise, the sequence diverges.
To find a sequence limit, first simplify the dominant behavior as n grows. For rational expressions, divide by the highest power of n; for exponential terms, compare growth rates; and for oscillating or bounded expressions, look for a useful squeeze. For example, a_n = (3n + 1)/(2n - 5) converges to 3/2 because the lower-order terms vanish after division by n.
The Monotone Convergence Theorem gives a powerful shortcut: every sequence that is both monotone and bounded converges. An increasing sequence needs an upper bound, while a decreasing sequence needs a lower bound. This theorem proves convergence even when finding the exact limit requires a separate algebraic step.