When a Test Does Not Decide
A convergence test is useful only when its conclusion follows from the test’s hypotheses and result. Some tests are decisive in one case and silent in another. In particular, a ratio-test limit of \(1\) does not tell us whether a series converges or diverges.
The right response is not to repeat the inconclusive test or to guess from a few terms. Instead, look for a different feature of the series: a familiar comparison, a function suitable for integration, or alternating signs with decreasing term sizes. The goal is to match the next test to the structure the first test did not capture.
What the Main Tests Can Establish
The following statements summarize the tests used in this workshop. A test’s conditions matter: a valid numerical limit is not enough if the required positivity, monotonicity, or other hypotheses are missing.
- Ratio test: For a series \(\sum a_n\), if \(L=\lim_{n\to\infty}|a_{n+1}/a_n|\) exists, then \(L<1\) implies absolute convergence, while \(L>1\) implies divergence. If \(L=1\), the test gives no conclusion.
- Direct comparison: For nonnegative terms, \(0\leq a_n\leq b_n\) and convergence of \(\sum b_n\) imply convergence of \(\sum a_n\). If \(0\leq b_n\leq a_n\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.
- Limit comparison: If \(a_n,b_n>0\) and \(\lim_{n\to\infty}a_n/b_n=c\), where \(0<c<\infty\), then \(\sum a_n\) and \(\sum b_n\) have the same convergence behavior.
- Integral test: If \(a_n=f(n)\), where \(f\) is positive, continuous, and decreasing for \(x\geq N\), then \(\sum_{n=N}^{\infty}a_n\) and \(\int_N^\infty f(x)\,dx\) either both converge or both diverge.
- Alternating-series test: If \(b_n\geq0\), \(b_n\) decreases to \(0\), then \(\sum (-1)^{n+1}b_n\) converges. This alone does not establish absolute convergence.
For positive series, familiar benchmarks include the geometric series and the \(p\)-series \(\sum 1/n^p\): the latter converges for \(p>1\) and diverges for \(p\leq1\). These benchmarks often make comparison more efficient than trying to extract a conclusion from a limit equal to \(1\).
Why an Inconclusive Result Must Stay Inconclusive
The ratio test compares neighboring terms. When their ratio approaches \(1\), that information is too coarse to distinguish series with different outcomes. For example, both \(1/n\) and \(1/n^2\) have consecutive-term ratios approaching \(1\), but their series behave differently.
The first series diverges, while the second converges. Therefore, a ratio limit of \(1\) cannot distinguish convergence from divergence. The proof of the ratio test’s usual conclusions depends on obtaining a ratio eventually bounded strictly below \(1\), or strictly above \(1\); a limit of \(1\) supplies neither bound.
Worked Example: Ratio Test Gives One, Comparison Decides
Worked Example: Compare with a Convergent \(p\)-Series
Determine whether \(\sum_{n=1}^{\infty}\frac{n}{n^3+1}\) converges.
First try the ratio test. With \(a_n=\frac{n}{n^3+1}\),
The numerator and denominator are both degree four in \(n\), with leading coefficient \(1\), so the ratio tends to \(1\). The ratio test is inconclusive.
For \(n\geq1\), the denominator \(n^3+1\) is at least \(n^3\), so
The benchmark series \(\sum 1/n^2\) converges because it is a \(p\)-series with \(p=2>1\). Direct comparison therefore proves that the given series converges.
Check: The comparison inequality holds for every \(n\geq1\), and the benchmark converges. Thus the comparison test’s hypotheses are satisfied; no conclusion was drawn from the ratio limit of \(1\).
Worked Example: Switch to the Integral Test
Worked Example: A Logarithmic Denominator
Determine whether \(\sum_{n=2}^{\infty}\frac{1}{n\ln n}\) converges.
Let \(a_n=\frac{1}{n\ln n}\). The ratio test gives
so it does not decide the question. Instead, take \(f(x)=\frac{1}{x\ln x}\) for \(x\geq2\). This function is positive and continuous there. Its derivative is
so it is decreasing. The integral test applies. Using \(u=\ln x\), \(du=dx/x\),
As \(R\to\infty\), \(\ln(\ln R)\to\infty\), so the improper integral diverges. The series therefore diverges.
Check: An antiderivative is \(\ln(\ln x)\), since its derivative is \(1/(x\ln x)\). The integral is unbounded, and the function satisfies all the integral-test hypotheses.
Worked Example: Alternation Can Rescue a Series
Worked Example: Conditional Convergence of an Alternating Series
Determine whether \(\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt n}\) converges, and decide whether it converges absolutely.
The ratio test on the absolute term \(b_n=1/\sqrt n\) gives
That test is inconclusive for absolute convergence. But \(b_n>0\), \(b_n\) decreases because \(1/\sqrt{n+1}<1/\sqrt n\), and \(b_n\to0\). The alternating-series test therefore proves that the signed series converges.
To test absolute convergence, consider \(\sum |(-1)^{n+1}/\sqrt n|=\sum 1/\sqrt n\). This is a \(p\)-series with \(p=1/2\leq1\), so it diverges. Hence the signed series converges, but not absolutely: it converges conditionally.
Check: The term magnitudes decrease to zero, as required for the alternating-series test. The absolute-value series is exactly the divergent \(p\)-series with \(p=1/2\), confirming conditional convergence rather than absolute convergence.
Worked Example: Use Limit Comparison for a Rational Term
Worked Example: Compare with the Harmonic Series
Determine whether \(\sum_{n=1}^{\infty}\frac{3n+2}{n^2+4n+1}\) converges.
The numerator has degree one and the denominator degree two, suggesting terms of order \(1/n\). Let \(a_n=\frac{3n+2}{n^2+4n+1}\) and \(b_n=1/n\). Both are positive for \(n\geq1\). Compute the limit comparison:
Since \(0<3<\infty\), limit comparison says the two series have the same behavior. The harmonic series \(\sum 1/n\) diverges, so the given series diverges.
Check: The terms are positive, the comparison limit exists and equals the finite positive number \(3\), and the benchmark diverges. Thus all conditions for limit comparison are met.
A Practical Switching Process
A deliberate sequence of decisions helps prevent an inconclusive result from being mistaken for a final answer.
Confirm that the terms tend to zero when necessary, and note positivity, alternating signs, factorials, logarithms, or dominant powers.
Try the ratio test for factorial or exponential behavior; try comparison for terms resembling a geometric or \(p\)-series benchmark; consider the integral test for a positive, decreasing function-sampled sequence.
If a test is inconclusive, keep the question open. Do not infer convergence or divergence from a limit equal to \(1\).
Choose another test, check its hypotheses, and state exactly what it establishes. For alternating series, separately test absolute convergence if that distinction matters.
The Root Test as a Related Diagnostic
The root test examines the \(n\)th root of the term rather than the ratio of consecutive terms. If \(L=\lim_{n\to\infty}\sqrt[n]{|a_n|}\) exists, then \(L<1\) implies absolute convergence and \(L>1\) implies divergence; \(L=1\) is inconclusive. It can be useful when terms are raised to the \(n\)th power, but it is not a universal remedy: a root-test limit of \(1\) also requires switching methods.
Common Pitfall: Treating One as a Verdict
A common incorrect calculation is: “The ratio-test limit is \(1\), so the series converges.” The conclusion does not follow. For \(\sum 1/n\), the ratio limit is \(1\), yet the series diverges. For \(\sum 1/n^2\), the ratio limit is also \(1\), yet the series converges.
The correction is to write “the ratio test is inconclusive” and then select a test that can decide the series. The harmonic-series benchmark settles the first example; the \(p\)-series benchmark with \(p=2\) settles the second. The limit itself establishes neither outcome.
Check Your Understanding
For each question, identify what the proposed test does establish and what further reasoning is needed.
- If the ratio-test limit is \(1\), what can you conclude about convergence?
- What comparison benchmark would you try for positive terms that behave like \(1/n^2\)?
- State the positivity, continuity, and monotonicity conditions needed to use the integral test.
- What two facts about \(b_n\) are required for the alternating-series test?
- If an alternating series converges by that test, does it necessarily converge absolutely? Explain what additional series you would test.
