Nth Term Test Calc 2 Calculator

Nth Term Test for Divergence:

\[ \text{If } \lim_{n \to \infty} a_n \neq 0 \text{ or does not exist, then } \sum_{n=1}^{\infty} a_n \text{ diverges} \]

Result:

Interpretation:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Nth Term Test?

The Nth Term Test (also known as the Divergence Test) is a fundamental test in calculus used to determine if an infinite series diverges. If the limit of the nth term does not approach zero as n approaches infinity, the series must diverge.

2. How Does the Test Work?

The test uses the following principle:

\[ \text{If } \lim_{n \to \infty} a_n \neq 0 \text{ or does not exist, then } \sum_{n=1}^{\infty} a_n \text{ diverges} \]

Where:

\( a_n \) — The nth term of the series
\( \lim_{n \to \infty} a_n \) — The limit of the nth term as n approaches infinity
\( \sum_{n=1}^{\infty} a_n \) — The infinite series being tested

Note: If the limit equals zero, the test is inconclusive - the series may either converge or diverge, and additional tests are needed.

3. Importance of the Test

Details: The Nth Term Test is often the first test applied when analyzing series convergence because it can quickly identify many divergent series without needing more complex tests.

4. Using the Calculator

Tips: Enter the expression for the nth term of your series. Use standard mathematical notation with 'n' as the variable (e.g., "1/n", "n^2/(n+1)", "sin(n)/n").

5. Frequently Asked Questions (FAQ)

Q1: What if the limit equals zero?
A: If the limit equals zero, the test is inconclusive. The series may converge (like ∑1/n²) or diverge (like ∑1/n). Additional convergence tests are needed.

Q2: Can this test prove convergence?
A: No, the Nth Term Test can only prove divergence. It cannot prove that a series converges, even if the limit is zero.

Q3: What types of expressions can I test?
A: You can test rational functions, trigonometric functions, exponential functions, and any other expressions where the limit as n→∞ can be determined.

Q4: Are there series where this test fails?
A: The test works for all series, but it's inconclusive when the limit is zero. Some oscillating series (like ∑(-1)ⁿ) clearly diverge even though the terms don't approach a single value.

Q5: How accurate is this calculator?
A: The calculator provides results based on standard limit evaluation techniques. For complex expressions, manual verification may be recommended.