An infinite series diverges if it does not approach a finite limit. Here's how to show that an infinite series diverges, incorporating insights from the provided video reference.
Understanding Divergence
A series, which is the sum of an infinite sequence of terms, is said to diverge if the sum does not approach a single, finite value. The provided YouTube video, "Convergence and Divergence - Introduction to Series," emphasizes this key concept.
Methods to Demonstrate Divergence
There are several methods for showing that an infinite series diverges, and we will use the nth Term Test for Divergence which is mentioned in the video.
The nth Term Test for Divergence
This is the first test to consider. It is stated as: If the limit of the sequence of terms (an) as n approaches infinity is not zero, then the series Σan diverges.
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Step 1: Identify the term (an): Determine the formula for the nth term of the series you are analyzing.
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Step 2: Compute the Limit: Find the limit of the term an as n approaches infinity.
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Step 3: Draw Conclusion: If the limit is not zero, the series diverges. If the limit is zero, the test is inconclusive. You then have to try another method to determine convergence or divergence.
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Example: The video uses the series where an = 2n.
- The limit of 2n as n approaches infinity is infinity, which is not zero.
- Therefore, the series Σ2n diverges.
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Another Example: The harmonic series Σ(1/n).
- The limit of 1/n as n approaches infinity is 0.
- This test fails and we need a different approach. The harmonic series diverges using the Integral Test for example.
Other Tests
While the nth term test is an initial check for divergence, other tests exist and might be necessary if the nth Term Test is inconclusive:
- Comparison Test: Compares your series to another known series to determine convergence/divergence.
- Limit Comparison Test: Compares your series to another known series using limits.
- Integral Test: Relates your series to an integral to see if it converges or diverges.
- Ratio Test: Tests the ratio of the terms of a series.
Key Takeaways
- The nth Term Test for Divergence is a primary method to show that an infinite series diverges.
- If the limit of the terms as n approaches infinity is anything other than zero, the series diverges.
- If the limit of the terms is zero, the test is inconclusive, and you'll need a different method to show divergence.
In summary, showing that an infinite series diverges involves demonstrating that the sum does not approach a finite limit. This often begins by applying the nth Term Test for Divergence.
