Series Convergence Calculator
Test if an infinite series converges or diverges using ratio, root, and comparison tests.
Use n as the index variable. Supports ^, *, /, sin, exp, log
What Is the Series Convergence Calculator?
The Series Convergence Calculator tests an infinite series Σ aₙ for convergence or divergence using three sequential tests: the Divergence Test, Ratio Test, and Root Test. For p-series (1/nᵖ patterns), it applies the p-Series Test directly. The calculator shows the first 15 terms, partial sums, and cumulative sums at n=100 and n=1000 to visualize convergence behavior.
- ›Enter the general term aₙ as a function of n (e.g. 1/n^2, 0.5^n, (-1)^n/n)
- ›Tests are applied in order: if Divergence Test is conclusive, remaining tests are skipped
- ›Partial sum table shows the first 15 individual terms and running totals
- ›n=1000 partial sum reveals whether the series converges or grows without bound
Formula
Convergence Tests
Divergence Test
If lim aₙ ≠ 0 → diverges
Ratio Test
L = lim |aₙ₊₁/aₙ|; L<1→converges, L>1→diverges
Root Test
L = lim |aₙ|^(1/n); same L rule as ratio
p-Series
Σ 1/nᵖ: converges if p>1, diverges if p≤1
Geometric
Σ rⁿ: converges if |r|<1, sum = a/(1−r)
Alternating
Σ (−1)ⁿbₙ: converges if bₙ↓0 (Leibniz test)
How to Use
- 1Enter the general term aₙ as an expression in n (the index variable)
- 2Set the starting index n (usually 1; use 0 for geometric series 0.5^n)
- 3Click Test Convergence, the verdict and all test results appear
- 4Review each test: Divergence, p-Series (if applicable), Ratio, and Root
- 5The partial sum table shows the first 15 terms and their running total
- 6Use the preset buttons to test classic series: 1/n², harmonic, geometric, alternating
Example Calculation
Σ 1/n² (Basel problem, starts at n=1):
p-Series test: p = 2 > 1 → CONVERGES
Ratio test: L = (1/(n+1)²) / (1/n²) = n²/(n+1)² → 1 (inconclusive)
Root test: L = (1/n²)^(1/n) → 1 (inconclusive)
Verdict: CONVERGES (confirmed by p-Series test)
Sum = π²/6 ≈ 1.6449340688... (Euler, 1735)
Partial sum n=100: 1.63498
Partial sum n=1000: 1.64393
The Harmonic Series diverges
Σ 1/n is the famous harmonic series. Despite aₙ → 0, the series diverges. This is the classic example showing that aₙ → 0 is necessary but not sufficient for convergence. The partial sums grow without bound, just very slowly.
Understanding Series Convergence
Convergence Test Decision Guide
| Series Type | Best Test | Verdict Condition | Example |
|---|---|---|---|
| Geometric rⁿ | Ratio Test | |r| < 1 | Σ (1/2)ⁿ → converges |
| p-series 1/nᵖ | p-Series Test | p > 1 | Σ 1/n² → converges |
| Factorial n! | Ratio Test | L < 1 | Σ n!/10ⁿ → diverges |
| Alternating (−1)ⁿbₙ | Leibniz/Alt. | bₙ ↓ 0 | Σ (−1)ⁿ/n → converges |
| Obvious divergence | Divergence | aₙ ↛ 0 | Σ n/(n+1) → diverges |
| Comparison | Comparison | Bound by known | Σ 1/(n²+1) ~ Σ 1/n² |
Frequently Asked Questions
What is an infinite series?
The key question for every infinite series is: does the sum approach a finite value, or does it grow without bound (or oscillate)?
- ›Convergent: Σ (1/2)ⁿ = 1 + 0.5 + 0.25 + ⋯ = 2 (geometric series)
- ›Divergent: Σ 1/n = 1 + 0.5 + 0.33 + ⋯ → ∞ (harmonic series)
- ›Conditionally convergent: Σ (−1)ⁿ/n → ln(2) (alternating harmonic)
- ›Partial sum Sₙ = a₁ + ⋯ + aₙ; series converges if lim Sₙ is finite
What is the Ratio Test?
The Ratio Test is especially powerful for series involving factorials and exponentials. It compares each term to the previous by computing the limiting ratio.
- ›L < 1: converges absolutely (terms shrink fast enough)
- ›L > 1: diverges (terms grow or don't shrink fast enough)
- ›L = 1: inconclusive, try p-Series, Root test, or integral test
- ›Best for: factorials (n!), exponentials (rⁿ), power series
What is the Divergence Test?
The Divergence Test is the first and fastest check. It can only prove divergence, it cannot prove convergence. Apply it first before any other test.
- ›lim aₙ ≠ 0 → series DIVERGES (definitive)
- ›lim aₙ = 0 → inconclusive (further tests needed)
- ›Classic example: Σ n/(n+1) diverges because aₙ → 1 ≠ 0
- ›Harmonic series: aₙ = 1/n → 0, but series still diverges (test inconclusive)
What is a p-series and when does it converge?
The p-series test is one of the most applied convergence tests in calculus because many series reduce to this form after simplification.
- ›p=1: Σ 1/n (harmonic), diverges, classic result
- ›p=2: Σ 1/n², converges to π²/6 ≈ 1.6449 (Basel problem)
- ›p=3: Σ 1/n³, converges to ζ(3) ≈ 1.202 (Apéry's constant)
- ›p=1/2: Σ 1/√n, diverges (p ≤ 1)
What is absolute vs. conditional convergence?
The alternating harmonic series Σ (−1)ⁿ/n converges conditionally, the series sums to ln(2) but the series of absolute values (harmonic) diverges.
- ›Absolute convergence: Σ |aₙ| converges (stronger property)
- ›Conditional convergence: Σ aₙ converges but Σ |aₙ| diverges
- ›Riemann rearrangement theorem: any conditionally convergent series can be rearranged to sum to any value!
- ›Absolute convergence implies convergence; the converse is false
What is a geometric series and what is its sum?
The geometric series is the most fundamental series in mathematics, appearing in finance (present value of annuities), probability, and physics.
- ›Σ (1/2)ⁿ from n=0: a=1, r=0.5 → sum = 1/(1−0.5) = 2
- ›Σ (1/3)ⁿ from n=0: sum = 1/(1−1/3) = 3/2
- ›Σ (0.9)ⁿ from n=0: sum = 1/(1−0.9) = 10
- ›Compound interest as geometric series: loan payments, bond pricing, perpetuities
When does the Ratio Test fail?
Many important series are boundary cases where L = 1, which is why multiple tests are available. This calculator applies the Ratio Test but also checks p-Series directly.
- ›1/nᵖ: ratio = (n/(n+1))ᵖ → 1, ratio test inconclusive
- ›1/(n ln n): diverges (integral test confirms), ratio test gives L=1
- ›In these cases: p-Series test or integral test are more informative
- ›Comparison test: compare to a known series (Σ 1/n² is a standard benchmark)