Compare custom positive-term models against classic benchmark series. Review limits, ratios, and verdict logic instantly. Export polished results for lessons, homework, audits, and revision.
Model positive-term series in the form c·rn/((n+s)p(ln(n+L))q). Then compare the tested series against a benchmark with either a direct or limit comparison approach.
| Scenario | Tested model | Benchmark model | Expected verdict |
|---|---|---|---|
| p-series stronger decay | 1 / n^2.2 | 1 / n^2 | Convergent |
| Borderline logarithmic case | 1 / (n(ln n)^1.4) | 1 / (n(ln n)^1.2) | Convergent |
| Harmonic-style divergence | 3 / (n + 1) | 1 / n | Divergent |
| Geometric dominance | 0.7^n / n | 0.8^n | Convergent |
This calculator samples positive-term models of the form:
a_n = c_a · (r_a)^n / ((n + s_a)^p_a (ln(n + L_a))^q_a) b_n = c_b · (r_b)^n / ((n + s_b)^p_b (ln(n + L_b))^q_b)For direct comparison, the calculator inspects whether sampled tail terms suggest a_n ≤ b_n or a_n ≥ b_n. For limit comparison, it samples a_n / b_n and checks whether the tail approaches a finite positive constant, zero, or grows without bound.
The benchmark series is classified using standard positive-term results. If r < 1, exponential decay guarantees convergence. If r = 1, the decision falls back to p-series and logarithmic criteria. When p > 1, the benchmark converges. When p = 1, convergence requires q > 1.
This tool is strongest for positive-term series with asymptotic behavior that matches common comparison families. It provides a guided estimate, not a symbolic proof for every possible series.
It compares a positive-term model against a benchmark series. Then it uses direct or limit comparison logic to estimate whether the tested infinite series converges or diverges.
Use limit comparison when both series have similar long-run structure. It works especially well when their ratio approaches a finite positive constant as n becomes large.
Choose direct comparison when you know one series is eventually smaller or larger than another benchmark. That inequality can immediately prove convergence or divergence for positive terms.
Standard comparison tests assume nonnegative terms. Negative or sign-changing terms usually require different tools, such as alternating series tests, absolute convergence checks, or rearrangement arguments.
Yes. The model includes a logarithmic factor (ln(n + L))^q, so it can represent cases like 1/(n(ln n)^q), which are common in comparison-test exercises.
This page estimates tail behavior numerically. Sampling helps visualize trends, but a formal proof still depends on establishing the eventual inequality or the exact limit analytically.
Yes, for positive-term series. If a_n/b_n approaches a nonzero finite constant, both series converge together or diverge together by the limit comparison test.
Yes. Export the summary and sampled data as CSV, or open the printable report and save it as a PDF from your browser.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.