Nth Term Test for Divergence

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Nth Term Test for Divergence

The Nth Term Test for Divergence is an essential tool for identifying whether an infinite series diverges. Here's a concise guide based on your detailed content:

Purpose:
To check if an infinite series diverges by examining the behavior of its general term a​n as n→∞.

Test Statement:

Why This Test Works:

  • For a series to converge, the general term an must approach 0 as n→∞.

  • If an does not approach 0, the series cannot converge, because the sum of infinitely large terms cannot settle to a finite value.


Steps to Apply:

  1. Identify the General Term: Write the formula for an​.

  2. Compute the Limit: Use algebraic techniques or calculus to evaluate lim⁡n→∞an​.

  3. Interpret the Result

Key Limitations:

  • Divergence Only: This test cannot prove convergence. A limit of zero is necessary for convergence but not sufficient.

  • Further Testing Required: When lim⁡n→∞an= 0, use additional tests (e.g., Comparison Test, Ratio Test, Integral Test).


Examples:

Example 1: Divergence

Series:

Example 2: Inconclusive

Series:

Common Misunderstandings:

Conclusion:

The Nth Term Test is a quick first step in determining whether an infinite series diverges. It simplifies analysis but must be supplemented with other tests for a complete understanding of series behavior.