Sequences Versus Series

A sequence a₁, a₂, a₃, ... is just a numbered list. Asking for its limit means asking where the individual terms end up as n gets huge. A series takes that same list and adds it up: a₁ + a₂ + a₃ + ... The terms of a sequence can shrink toward some fixed number while the running sum keeps climbing forever, because even tiny terms added infinitely many times can pile up without bound. That is why checking a sequence's limit and checking a series' behavior are genuinely different questions, even though they use the same list of numbers.

To find the limit of a sequence given as a ratio of polynomials in n, divide every term, top and bottom, by the highest power of n that appears. Each piece with n in the denominator shrinks to 0, leaving behind whatever constants survive. For (3n+1)/(2n-5), dividing by n gives (3 + 1/n)/(2 - 5/n), which heads to 3/2. Sequences built from exponentials or factorials often need a different tool: recognize that exponential growth, like 2ⁿ, eventually outpaces any fixed power of n, so ratios like n²/2ⁿ collapse to 0.

This is exactly the limit-at-infinity work from early in the course, just relabeled: n replaces x, and n is only allowed to take whole-number values marching upward. Every technique from back then, dividing by the leading power, comparing growth rates, even L'Hopital's rule treating n as a continuous variable, carries over directly. Getting comfortable with sequence limits here sets up everything that follows: a series converges only when its sequence of partial sums has a limit, so this skill is the foundation under the whole unit.