1. What is the Nth Term Test for Divergence?

The nth term test for divergence is a simple test that can determine if an infinite series diverges by examining the limit of its sequence of terms.

2. How Does the Test Work?

The test states:

Important Notes:

• If the limit is not zero, the series definitely diverges
• If the limit is zero, the test is inconclusive (the series may converge or diverge)
• This test can only prove divergence, never convergence

3. When to Use This Test

Details: Use this test as a first check when examining any series. It's quick and can immediately identify some divergent series.

4. Using the Calculator

Tips: Enter the general term of your sequence (aₙ). For best results, use simple rational expressions in n.

5. Frequently Asked Questions (FAQ)

Q1: Can this test prove convergence?
A: No, it can only prove divergence. If lim aₙ = 0, you must use other tests to determine convergence.

Q2: What are common examples where this test applies?
A: The test shows divergence for series like Σ(1), Σ(n), Σ(n/(n+1)), etc.

Q3: What if the limit doesn't exist?
A: If lim aₙ doesn't exist (and isn't ±∞), then the series diverges.

Q4: Are there series where the terms go to zero but diverge?
A: Yes, the harmonic series Σ(1/n) is the classic example - terms →0 but the series diverges.

Q5: How is this different from the ratio test?
A: The ratio test examines the limit of ratios of terms and can sometimes prove convergence.