Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]
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14. Sequences & Series
Convergence Tests
3:13 minutesProblem 10.6.65c
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. A series that converges conditionally must converge.
Verified step by step guidance1
Recall the definition of conditional convergence: A series \( \sum a_n \) converges conditionally if it converges, but does not converge absolutely. That is, \( \sum a_n \) converges, but \( \sum |a_n| \) diverges.
The statement says: "A series that converges conditionally must converge." By the definition of conditional convergence, the series does converge (but not absolutely). So the statement is true by definition.
To clarify, conditional convergence implies convergence of the original series, but not absolute convergence. This means the series converges, but the sum of the absolute values of its terms does not.
A classic example of a conditionally convergent series is the alternating harmonic series \( \sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n} \), which converges, but the harmonic series \( \sum_{n=1}^\infty \frac{1}{n} \) diverges, so it is not absolutely convergent.
Therefore, the statement is true because conditional convergence by definition requires the series to converge (just not absolutely).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conditional Convergence
A series converges conditionally if it converges but does not converge absolutely. This means the series converges when considering the terms with their signs, but the series of absolute values diverges. An example is the alternating harmonic series.
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Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series, and it is a stronger form of convergence than conditional convergence.
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Convergence of Series
A series converges if the sequence of its partial sums approaches a finite limit. Conditional convergence implies the series converges, but not absolutely. Therefore, any conditionally convergent series must converge, but not all convergent series are conditionally convergent.
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