Determine If The Following Series Converges or Diverges Calculator

Introduction to Series Convergence

An infinite series is a sum of infinitely many terms. A series converges if the sum approaches a finite value as the number of terms increases. Otherwise, it diverges.

There are several tests to determine series convergence, including:

• Divergence Test
• Geometric Series Test
• Comparison Test
• Ratio Test
• Root Test
• Integral Test

This calculator implements these tests to analyze your series.

Convergence Tests

Divergence Test

The simplest test: if the limit of the general term does not approach zero, the series diverges.

Geometric Series Test

A series of the form Σ (a * r^n) converges if |r| < 1.

Comparison Test

If 0 ≤ aₙ ≤ bₙ and Σ bₙ converges, then Σ aₙ converges.

Ratio Test

If lim (|aₙ₊₁/aₙ|) = L, then:

• L < 1: converges absolutely
• L > 1: diverges
• L = 1: inconclusive

Root Test

If lim (√n |aₙ|) = L, then:

• L < 1: converges
• L > 1: diverges
• L = 1: inconclusive

Integral Test

If f(n) = aₙ is continuous, positive, and decreasing, then Σ aₙ and ∫ f(x) dx have the same convergence.

Worked Examples

Example 1: Geometric Series

Series: Σ (1/2)^n

Analysis: This is a geometric series with r = 1/2. Since |r| < 1, it converges to 2.

Example 2: p-Series

Series: Σ 1/n^2

Analysis: This is a p-series with p = 2 > 1, so it converges.

Example 3: Harmonic Series

Series: Σ 1/n

Analysis: The harmonic series diverges because the terms do not approach zero.

Interpreting Results

When the calculator determines a series converges, it provides the approximate sum. For divergent series, it explains why the sum does not approach a finite value.

Remember that some tests may be inconclusive, requiring additional analysis.