1. What is the Nth Term Test?

The Nth Term Test (also called the Divergence Test) is a preliminary test for series convergence. It states that if the limit of a sequence's terms doesn't approach zero, then the series must diverge.

2. How Does the Test Work?

The test uses the following logic:

Important Notes:

• The test can only prove divergence, not convergence
• If lim aₙ = 0, the test is inconclusive (series may converge or diverge)
• This is always the first test to apply when analyzing series

3. Importance of the Nth Term Test

Details: This simple test can quickly identify many divergent series without needing more complex tests. It's particularly useful for series with obvious non-zero limits.

4. Using the Calculator

Tips: Enter the general term of your sequence (aₙ) using 'n' as the variable. The calculator recognizes common patterns like 1/n, n/(n+1), (-1)ⁿ/n, etc.

5. Frequently Asked Questions (FAQ)

Q1: If the limit is zero, does that mean the series converges?
A: No! The test is inconclusive when the limit is zero. The series may converge (e.g., ∑1/n²) or diverge (e.g., ∑1/n).

Q2: What if the limit doesn't exist?
A: If the limit doesn't exist (not even as ±∞), then the series diverges by the nth term test.

Q3: Can this test prove convergence?
A: No, it can only prove divergence. You need other tests (comparison, ratio, integral, etc.) to prove convergence.

Q4: What about alternating series?
A: The test works the same way - if the non-alternating part doesn't approach zero, the series diverges.

Q5: How precise does my input need to be?
A: The calculator recognizes common patterns but may not understand complex expressions. For advanced sequences, manual analysis is needed.