Convergence tests

Remarks

Example: the property "there are $q < 1$ and $N$ with $\abs{a_{n+1}} \le q \abs{a_n}$ for all $n \ge N$" is a convergence test. Indeed $\abs{a_n} \le \abs{a_N} q^{n - N}$ for $n \ge N$, and the comparison test with a geometric series gives convergence. Non-example: the property "$a_n \to 0$" is not a convergence test. The terms $1/n$ of the harmonic series tend to $0$, but the series diverges. Its negation "$a_n \not\to 0$" is a divergence test. A test is inconclusive on a series when the series has neither the property of the test nor the property of its companion divergence test.