The interval of convergence of an infinite series is determined by finding the range of values for which the series converges. This is often done using the ratio test.
Understanding the Ratio Test
The ratio test for absolute convergence is a powerful tool for determining if a series converges. It states:
- If
lim n → ∞ | aₙ₊₁ / aₙ | < 1, the series ∑ aₙ converges absolutely. - If
lim n → ∞ | aₙ₊₁ / aₙ | > 1, or the limit does not exist, the series ∑ aₙ diverges. - If
lim n → ∞ | aₙ₊₁ / aₙ | = 1, the test is inconclusive.
The reference information states: "The interval of convergence can be found using the ratio test for absolute convergence. Ratio Test for Absolute Convergence: The ratio test for absolute convergence states that a series ∑ n = 0 ∞ a n converges absolutely (and thus converges) if lim n → ∞ | a n + 1 a n | < 1."
Steps to find the Interval of Convergence
Here's how to use the ratio test to find the interval of convergence:
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Set up the Ratio: Start with the given series ∑aₙ. Find the next term in the sequence, aₙ₊₁. Form the ratio |aₙ₊₁ / aₙ|.
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Evaluate the Limit: Calculate the limit of this ratio as n approaches infinity: lim n → ∞ |aₙ₊₁ / aₙ|.
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Apply the Ratio Test: Based on the limit, use these rules:
- Converges Absolutely: If the limit is less than 1, the series converges absolutely, and thus it converges.
- Diverges: If the limit is greater than 1 or does not exist, the series diverges.
- Inconclusive: If the limit is equal to 1, the ratio test fails. You'll need to use another test.
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Determine the Interval: After applying the ratio test, you will obtain an inequality, typically with x in it. By solving for x, you'll find the interval where the series converges.
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Check Endpoints: The ratio test provides the interior of the interval. You must manually test the endpoints to determine if the series converges or diverges at each endpoint.
Example
Let's look at an example:
Find the interval of convergence of the series ∑ (xⁿ / n).
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Ratio Setup:
- aₙ = xⁿ / n
- aₙ₊₁ = xⁿ⁺¹ / (n + 1)
- Ratio: | aₙ₊₁ / aₙ | = | (xⁿ⁺¹ / (n + 1)) / (xⁿ / n) | = | xⁿ⁺¹ n / (xⁿ(n + 1)) | = | x n / (n + 1) |
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Limit Evaluation:
- lim n → ∞ | x n / (n + 1) | = | x | lim n → ∞ | n / (n + 1) | = | x | * 1 = | x |
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Applying Ratio Test:
- For convergence, | x | < 1 which implies -1 < x < 1.
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Check Endpoints:
- At x = 1, the series becomes ∑ (1 / n), the harmonic series which diverges.
- At x = -1, the series becomes ∑ ((-1)ⁿ / n), which converges by the alternating series test.
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Interval of Convergence:
- The interval of convergence is [-1, 1), meaning the series converges for all values of x greater than or equal to -1 and less than 1.
Summary
The ratio test helps you find the values of x for which a series converges, giving the interval of convergence. Remember to always check the series behavior at the interval's endpoints.
