The divergence test, also known as the nth-term test, is a useful tool in determining whether an infinite series diverges. You should use it when you want to quickly check if a series diverges.
Here's how to use the divergence test:
- Identify the series: Start by clearly defining the infinite series you're working with.
- Find the limit of the nth term: Calculate the limit of the sequence's terms as n approaches infinity.
- Apply the test: If the limit of the nth term is not equal to zero, then the series diverges. If the limit is zero, the test is inconclusive, and you'll need to use another test to determine convergence or divergence.
Example:
Consider the series: 1 + 2 + 3 + 4 + ...
- Identify the series: The series is the sum of all natural numbers.
- Find the limit of the nth term: The nth term is n, and the limit of n as n approaches infinity is infinity.
- Apply the test: Since the limit of the nth term is not zero, the series diverges.
Practical Insights:
- The divergence test is a quick and easy way to rule out convergence for some series.
- It's particularly useful when dealing with series where the terms don't decrease to zero.
- If the limit of the nth term is zero, the divergence test doesn't provide any information about the series' convergence or divergence. You'll need to use other tests like the ratio test, integral test, or comparison test.
In summary: The divergence test is a simple but powerful tool for determining divergence in infinite series. It's most effective when the limit of the nth term is not zero, providing a straightforward way to identify divergent series.
