Study series with ratio insights and alternating checks. Test convergence, divergence, or inconclusive outcomes confidently. Built for learners needing clear numeric evidence every time.
Use a positive base for exponential and factorial families. Conditional convergence appears only when the signed series converges while the absolute series fails.
| Example | Family | Base r | p | Alternating | Ratio Limit L | Expected Outcome |
|---|---|---|---|---|---|---|
| ∑ (0.5ⁿ / n) | Exponential Over Polynomial | 0.5 | 1 | No | 0.5 | Absolutely convergent |
| ∑ ((-1)ⁿ / n) | Pure p-Series | 1 | 1 | Yes | 1 | Conditionally convergent |
| ∑ (1 / n²) | Pure p-Series | 1 | 2 | No | 1 | Absolutely convergent |
| ∑ (2ⁿ / (n!·n)) | Factorial in Denominator | 2 | 1 | No | 0 | Absolutely convergent |
| ∑ (n! / 2ⁿ) | Factorial in Numerator | 2 | 0 | No | ∞ | Divergent |
The ratio test studies the limit L = lim |aₙ₊₁ / aₙ|.
Conditional convergence needs two facts. The original signed series must converge. The absolute version must diverge.
For the built-in families:
It measures how fast consecutive terms change in magnitude. The limit of |aₙ₊₁/aₙ| tells whether terms shrink quickly enough for absolute convergence.
It is inconclusive when the limit equals 1. In that case, another method must decide convergence, such as the p-series test or alternating series test.
A series is conditionally convergent when the signed series converges, but the series of absolute values diverges. Alternating harmonic series is the standard example.
Not usually. The ratio test mainly detects absolute convergence or divergence. Conditional convergence often appears only after the ratio test becomes inconclusive.
Factorials grow very quickly. When they appear in the denominator, they force terms to shrink rapidly, often making the ratio limit equal zero.
A factorial numerator usually grows faster than exponential and polynomial pieces. That growth prevents terms from shrinking sufficiently, so divergence is common.
It shows term magnitudes and partial sums. Together, they reveal whether terms decay and whether cumulative totals appear to stabilize.
No. It covers important structured families well. For unusual terms, the ratio idea still helps, but symbolic work may be needed separately.
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