What is series convergence?
An infinite series Σ aₙ converges if its partial sums approach a finite limit as n → ∞. If the partial sums grow without bound or oscillate without settling, the series diverges.
**Convergent example:** 1 + 1/2 + 1/4 + 1/8 + ... = 2 (geometric series, |r| < 1)
**Divergent example:** 1 + 1/2 + 1/3 + 1/4 + ... = ∞ (harmonic series)
Convergence tests covered
**Geometric Series Test** For Σ arⁿ: converges if |r| < 1, diverges if |r| ≥ 1. Sum = a/(1−r).
**p-Series Test** For Σ 1/nᵖ: converges if p > 1, diverges if p ≤ 1.
**Ratio Test (D'Alembert)** Compute L = lim|a_{n+1}/a_n|. - L < 1: absolutely convergent - L > 1: divergent - L = 1: inconclusive
**Root Test (Cauchy)** Compute L = lim|aₙ|^(1/n). - L < 1: absolutely convergent - L > 1: divergent - L = 1: inconclusive
**Alternating Series Test (Leibniz)** For Σ (-1)ⁿ bₙ: converges if bₙ is decreasing and bₙ → 0.
**Integral Test** If f(x) is positive and decreasing, Σ f(n) converges iff ∫₁^∞ f(x) dx converges. Used internally for p-series.
Absolute vs conditional convergence
A series converges **absolutely** if Σ |aₙ| converges. Absolute convergence implies convergence.
A series converges **conditionally** if Σ aₙ converges but Σ |aₙ| diverges.
**Example:** The alternating harmonic series Σ (-1)ⁿ/n converges conditionally. Its absolute value series Σ 1/n (harmonic) diverges.
Absolute convergence is stronger and implies the series can be rearranged without changing its sum (Riemann's rearrangement theorem doesn't apply).
Common series and their convergence
| Series | Form | Converges? | Sum | |--------|------|-----------|-----| | Geometric | Σ rⁿ, |r|<1 | Yes | 1/(1−r) | | Harmonic | Σ 1/n | No | ∞ | | Alt. Harmonic | Σ (-1)ⁿ/n | Yes (cond.) | ln(2) | | p-Series p>1 | Σ 1/nᵖ | Yes | ζ(p) | | Basel | Σ 1/n² | Yes | π²/6 | | Exponential | Σ xⁿ/n! | Yes | eˣ |
How to use
1. Select the series type from the dropdown 2. Enter any required parameters (ratio, exponent, limit value) 3. Click "Test Convergence" 4. Review the verdict and individual test results 5. Read the explanation for a plain-English summary
FAQs
Q: What is the ratio test? A: The ratio test computes L = lim|a_{n+1}/a_n|. If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, the test is inconclusive and another test must be used.
Q: What is the difference between convergence and absolute convergence? A: A series converges absolutely if the sum of the absolute values converges. Conditional convergence means the series converges but not absolutely. The alternating harmonic series is the classic example of conditional convergence.
Q: When is the ratio test inconclusive? A: When L = 1 exactly. This happens for p-series (Σ 1/nᵖ) of all types. For these, use the integral test or p-series test instead.
Q: What does the p-series test say about Σ 1/n²? A: Since p = 2 > 1, the series converges. The exact sum is π²/6 ≈ 1.6449 (the Basel problem, solved by Euler in 1734).
Q: Can I test any series with this tool? A: This tool covers the most common series types taught in calculus. For arbitrary series, enter the limit L for the ratio or root test, or check the alternating series conditions manually.
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