Test The Convergence of The Following Series Calculator with Steps

Introduction

An infinite series is the sum of an infinite sequence of numbers. Convergence refers to whether this sum approaches a finite limit as the number of terms increases. Testing series convergence is fundamental in calculus and analysis.

Common convergence tests include the Ratio Test, Root Test, Comparison Test, Integral Test, and Alternating Series Test. Each test has specific conditions that must be satisfied for the series to converge.

Common Convergence Tests

Ratio Test

The Ratio Test states that for a series ∑aₙ, if lim (n→∞) |aₙ₊₁/aₙ| = L, then:

• If L < 1, the series converges absolutely.
• If L > 1, the series diverges.
• If L = 1, the test is inconclusive.

Root Test

The Root Test examines lim (n→∞) √ⁿ|aₙ| = L. The series:

• Converges if L < 1.
• Diverges if L > 1.
• Is inconclusive if L = 1.

Comparison Test

If 0 ≤ aₙ ≤ bₙ for all n and ∑bₙ converges, then ∑aₙ also converges. If aₙ ≥ bₙ ≥ 0 and ∑bₙ diverges, then ∑aₙ diverges.

Integral Test

For a decreasing, continuous, and positive function f(x) ≥ 0, the series ∑f(n) converges if ∫ from 1 to ∞ f(x) dx converges.

Alternating Series Test

If a series alternates in sign and |aₙ₊₁| ≤ |aₙ| for all n, and lim (n→∞) aₙ = 0, then the series converges.

Using the Calculator

Our calculator applies the Ratio Test by default. Enter the general term of your series and click "Calculate" to determine convergence. The calculator provides step-by-step results and visualizations.

Worked Examples

Example 1: Convergent Series

Consider the series ∑(1/n²). Using the Ratio Test:

1. Compute lim (n→∞) |(1/(n+1)²)/(1/n²)| = lim (n→∞) n²/(n+1)² = 1.
2. Since L = 1, the Ratio Test is inconclusive.
3. Apply the Root Test: lim (n→∞) √ⁿ(1/n²) = lim (n→∞) 1/n = 0 < 1.
4. The series converges by the Root Test.

Example 2: Divergent Series

For the series ∑(1/n), the Ratio Test gives lim (n→∞) |(1/(n+1))/(1/n)| = 1. The Root Test gives lim (n→∞) √ⁿ(1/n) = 1. The series diverges by the Divergence Test (lim aₙ ≠ 0).

Interpreting Results

If the calculator indicates convergence, the series sum approaches a finite limit. If it shows divergence, the series does not approach a finite limit. Absolute convergence implies uniform convergence and is stronger than conditional convergence.