Limit Comparison Test

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

1. Verify eventual positivity.
2. Choose $b_n$ from dominant powers or a familiar benchmark.
3. Compute and simplify $a_n/b_n$.
4. Check that the limit is finite and strictly positive.
5. State the benchmark behavior and transfer it to the target.

Fully worked example

Graphical or geometric meaning

For large $n$, the target term is about three times the benchmark. Their partial tail sizes therefore differ by scale, not by whether they are finite. Early irregular terms do not affect this tail comparison.

Common mistakes and why they fail

Verification and reasonableness checks

• Estimate dominant powers before calculating the ratio.
• Confirm the limiting constant is neither zero nor infinite.
• Use direct inequalities as an independent tail check when simple.

A finite positive limit transfers behavior

For eventually positive terms, if $\lim a_n/b_n=L$ with $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ share convergence behavior. Choose $b_n$ from the dominant powers or factors of $a_n$, and compute the limit rather than merely claiming resemblance. Limits zero or infinity do not give the two-way conclusion, though a one-sided comparison may still work in the useful direction. If signs vary, use absolute values only when testing absolute convergence. A constant multiple does not affect convergence, which is exactly what a finite positive $L$ captures. Finish by naming the benchmark—often a $p$-series or geometric series—and stating why it converges or diverges.

Practice

1. Compare $1/(n^2+n)$ with $1/n^2$.
2. Compare $(2n+1)/(n^2+4)$ with $1/n$.
3. Does changing ten initial terms matter?

Connections and next steps

• Comparison Test
• Harmonic Series
• Ratio Test