Comparison Tests for Convergence BC only

With $0 \le a_n \le b_n$ eventually, only two of the four possible pairings conclude anything: smaller than a convergent series gives convergence, and larger than a divergent one gives divergence. The other two are empty rather than weak, and the figure makes that concrete, since $\frac{1}{2^{n}} \le \frac{1}{n}$ is true while $\sum \frac{1}{n}$ diverges and $\sum \frac{1}{2^{n}}$ converges anyway; applying the same useless pairing to $\frac{1}{2n} \le \frac{1}{n}$ gives a divergent series instead, so the pairing predicts nothing. A comparison helps only when it pushes the way you want to go. The stock of known series is geometric and $p$-series, and choosing from it means subtracting degrees for rational terms, letting an exponential dominate a polynomial, and dropping added constants.

The limit comparison test replaces the inequality with $L = \lim \frac{a_n}{b_n}$: when $L$ is finite and positive the two series do the same thing, a two-way conclusion needing no inequality at all. Its boundary cases are one-directional and are where the conclusion errors live, since $L = 0$ only transfers convergence downward and $L = \infty$ only transfers divergence upward, which are the same two useful pairings as before. Both tests require positive terms, so nothing alternating is eligible until 10.9. A complete answer names the comparison series, states its verdict with a reason, and then either proves the inequality in the needed direction or computes $L$ and says it is finite and positive; asserting that two series behave alike is the intuition these tests exist to replace.