Math101 · Independent Ontario learning library
Math101Limit Comparison Test
A rigorous, example-driven guide to limit comparison test, including hypotheses, method choice, verification, and practice.
The central idea
For positive sequences $a_n,b_n$, if $\lim_{n\to\infty}a_n/b_n=L$ with $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ have the same convergence behavior. Limits $0$ or $\infty$ give only one-way conclusions and are not the standard equivalence form.
Definitions, hypotheses, and notation
Dominant-term selection usually keeps only the highest power in numerator and denominator. For radicals, rationalization or exponent rewriting may reveal the benchmark. The test is often shorter than proving a global inequality, but it still depends on a computed limit and a benchmark whose behavior is already known.
If $L=0$ and the benchmark converges, direct comparison reasoning can still imply target convergence; if $L=\infty$ and the benchmark diverges, target divergence may follow. The clean 'same behavior' conclusion, however, requires $0<L<\infty$, so state precisely which version is being used.
Conceptual meaning
A finite positive ratio says the terms are asymptotically constant multiples of one another. Far enough out, each series bounds the other up to fixed factors, so their tails are simultaneously finite or infinite.
A dependable method and decision rule
- Verify eventual positivity.
- Choose $b_n$ from dominant powers or a familiar benchmark.
- Compute and simplify $a_n/b_n$.
- Check that the limit is finite and strictly positive.
- State the benchmark behavior and transfer it to the target.
