limit comparison test
The limit comparison test refines the comparison test by asking not whether one series is term-by-term smaller, but whether two series grow at essentially the same rate in the long run. If their terms are eventually proportional, the two series stand or fall together — you no longer need a clean inequality, only matching asymptotics.
Precisely, let a_n > 0 and b_n > 0 and suppose the limit L = lim (a_n / b_n) exists. If 0 < L < infinity, then sum a_n and sum b_n either both converge or both diverge. There are also one-sided variants: if L = 0 and sum b_n converges, then sum a_n converges; if L = infinity and sum b_n diverges, then sum a_n diverges. The finite-nonzero case is by far the most used.
Why this helps in practice: it spares you from constructing exact inequalities, which can be fiddly. You simply read off the dominant behaviour of a_n for large n, choose a b_n with the same dominant behaviour (typically a p-series), and compute the ratio's limit. The honest restriction is that the terms must be (eventually) positive, and a genuine limit of the ratio must exist.
For a_n = (2n + 1)/(n^3 + 5), take b_n = 1/n^2; then a_n / b_n -> 2, a finite nonzero limit. Since sum 1/n^2 converges, so does sum (2n + 1)/(n^3 + 5).
Matching asymptotics to a p-series settles convergence without inequalities.