Why the Ratio Is the Key Calculation
When a series has positive terms that combine several familiar patterns, its convergence can be hard to see directly. A useful approach is to compare each term with a simpler benchmark sequence, such as \(1/n^p\) or \(1/n\). The central calculation is the ratio of the given term to the benchmark term.
The ratio only helps if it is simplified correctly and interpreted with the right hypotheses. A finite positive limit gives a strong conclusion; a limit of zero or infinity gives only a one-sided conclusion. In some cases, the chosen benchmark gives no conclusion at all.
“Positive for all sufficiently large \(n\)” means that there is some index after which every term is positive. A finite number of earlier terms does not affect whether a series converges, so comparison tests may be applied to the tail.
What Each Type of Limit Tells You
The standard limit comparison theorem concerns a finite, strictly positive ratio limit. Its proof also explains why positivity and the benchmark’s behavior matter.
To see why, use the definition of a limit with tolerance \(c/2\). Since \(a_n/b_n\) tends to \(c\), eventually the ratio lies between \(c/2\) and \(3c/2\). Multiplying by the positive quantity \(b_n\) gives two constant-multiple bounds.
For sufficiently large \(n\), \(\left|a_n/b_n-c\right|<c/2\).
The inequality implies \(c/2<a_n/b_n<3c/2\). Since \(b_n>0\), it follows that \((c/2)b_n<a_n<(3c/2)b_n\).
Multiplying terms by a fixed positive constant does not change whether their series converges. The two bounds therefore give the same convergence behavior for the tails of \(\sum a_n\) and \(\sum b_n\).
This proof justifies using the ratio in either direction: when its limit is a finite positive number, the two sequences have the same series behavior. The conclusion does not extend unchanged to a limit of zero or infinity.
For the zero-limit conclusion, eventually \(a_n/b_n<1\), so \(0<a_n<b_n\). For the infinity-limit conclusion, eventually \(a_n/b_n>1\), so \(a_n>b_n>0\). These bounds point in the useful direction for the stated conclusions. A bound above by a divergent sequence, or below by a convergent sequence, generally does not decide convergence.
Worked Examples: Simplifying and Using Ratios
Worked Example: A small-angle term and a power benchmark
For \(n\ge1\), consider \(a_n=\sin(1/n)/n^2\). Choose \(b_n=1/n^3\). The benchmark series \(\sum 1/n^3\) converges, since it is a \(p\)-series with \(p=3>1\). We simplify the ratio:
As \(n\to\infty\), \(1/n\to0\). The small-angle limit \(\sin x/x\to1\) as \(x\to0\) therefore gives
The terms are positive because \(0<1/n\le1\), so \(\sin(1/n)>0\). The ratio limit is finite and strictly positive, and the benchmark series converges. By limit comparison, \(\sum \sin(1/n)/n^2\) converges as well.
As a check on the ratio calculation, dividing \(a_n\) by \(b_n\) cancels the factor \(n^2\) in the denominator and leaves \(n\sin(1/n)\), exactly the expression used in the limit.
Worked Example: A zero limit with a convergent benchmark
For \(n\ge1\), let \(a_n=1/[n^2(1+\ln(n+1))]\). Choose \(b_n=1/n^2\), whose series converges because \(p=2>1\). The ratio simplifies directly:
Because \(\ln(n+1)\to\infty\), the denominator tends to infinity, and hence
Both sequences are positive. Since the benchmark series converges and the ratio tends to zero, the zero-limit comparison implies that \(\sum a_n\) converges. Explicitly, the ratio is eventually less than \(1\), so \(0<a_n<b_n\) from some index onward; the comparison test then gives convergence.
The check is the same cancellation as in the ratio: the factor \(n^2\) disappears, leaving the reciprocal logarithmic factor. As that factor tends to zero, \(a_n\) is eventually smaller than the convergent benchmark.
Worked Example: An infinite limit with a divergent benchmark
For \(n\ge1\), let \(a_n=\ln(n+1)/n\), and choose the harmonic benchmark \(b_n=1/n\). The ratio is
Since \(\ln(n+1)\to\infty\), the ratio tends to infinity. The harmonic series \(\sum 1/n\) diverges. The infinity-limit comparison therefore implies that \(\sum \ln(n+1)/n\) diverges.
To check the direction of the conclusion, the ratio eventually exceeds \(1\). Thus \(a_n>b_n\) eventually. More concretely, for \(n\ge2\), \(n+1\ge3>e\), so \(\ln(n+1)>1\), giving \(\ln(n+1)/n>1/n\). The terms dominate the terms of a divergent series, as required.
Choosing a Benchmark That Can Answer the Question
A ratio is not useful merely because its limit exists. The benchmark must also have known series behavior, and the limit must point in a direction that supports a conclusion. Common benchmarks include \(1/n^p\), whose series converges for \(p>1\) and diverges for \(p\le1\), and geometric terms \(r^n\), whose series converges when \(|r|<1\).
A practical calculation can be organized into four steps:
Match the term’s dominant scale to a familiar sequence with known convergence behavior.
Write \(a_n/b_n\), keeping track of which term is the series under investigation and which is the benchmark.
Cancel common factors, divide by dominant powers, or rewrite the expression so the limiting behavior is visible.
Check positivity, the benchmark’s known behavior, and whether the limit is finite positive, zero, or infinite.
The order of the ratio matters. If a problem gives a complicated term and a benchmark, writing down \(a_n/b_n\) explicitly prevents accidentally calculating \(b_n/a_n\). Reversing the ratio changes zero to infinity and can reverse which comparison conclusion is available.
A Common Mistake: Treating Every Zero Limit as Convergence
Suppose \(a_n=1/n^2\) and choose the harmonic benchmark \(b_n=1/n\). Then \(a_n/b_n=1/n\to0\). It would be incorrect to conclude that \(\sum a_n\) diverges because \(\sum b_n\) diverges. The zero-limit result does not say that a smaller sequence inherits divergence from a larger divergent sequence.
In fact, this example has a better benchmark: take \(c_n=1/n^2\). Then
The benchmark series \(\sum c_n\) converges, so the finite-positive limit comparison theorem confirms that \(\sum a_n\) converges. The mistake was not in calculating the first ratio; it was in drawing a conclusion that the zero-limit comparison does not permit.
Why These Ratios Matter
The ratio method separates two tasks that are easy to confuse. First, simplify the algebra and find the limiting relative size of the terms. Second, interpret that limit using the benchmark’s known series behavior. A correct limit does not guarantee a correct conclusion unless both tasks are completed.
In particular, verify that the terms are positive eventually, that the ratio has been formed in the stated order, and that the benchmark is a standard series whose behavior is known. A finite positive limit gives equivalence of convergence behavior; a zero or infinite limit only supports the appropriate one-sided comparison.
Check Your Understanding
For each question, identify both the ratio limit and the conclusion that the comparison does or does not justify.
- If \(a_n/b_n\to 4\) and \(\sum b_n\) diverges, what can you conclude about \(\sum a_n\)?
- If \(a_n/b_n\to0\) and \(\sum b_n\) converges, which inequality eventually relates \(a_n\) and \(b_n\)?
- If \(a_n/b_n\to\infty\) but \(\sum b_n\) converges, does the ratio alone determine whether \(\sum a_n\) converges?
- For \(a_n=\ln(n+1)/n^2\) and \(b_n=1/n^2\), simplify \(a_n/b_n\) and find its limit.
- Why is a finite positive ratio limit stronger than a zero ratio limit for deciding whether two positive-term series have the same behavior?