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Chapter 9: Problem 7

Is it possible for a series of positive terms to converge conditionally? Explain.

Short Answer

Expert verified
Answer: No, a series with positive terms cannot converge conditionally, as it lacks the alternating signs needed for conditional convergence. It will either converge absolutely or not converge at all.

Step by step solution

01

Definition of Conditional Convergence and Absolute Convergence

Conditional convergence refers to a series that converges when the terms are alternately summed with a change of sign, but not when they are summed with the same sign. In other words, a series converges conditionally if it converges but does not converge absolutely. On the other hand, absolute convergence means that a series converges when the absolute values of its terms are summed. If a series converges absolutely, then it converges as well.
02

Testing for Convergence

To determine if a series converges, we can apply different convergence tests. Commonly used tests include the Ratio Test, Root Test, Comparison Test, Limit Comparison Test, and Alternating Series Test. It's important to note that the convergence tests can only be applied to series with non-negative terms.
03

Series with Positive Terms

If a series has positive terms, it is not possible for it to converge conditionally. Conditional convergence involves alternating signs in the series terms. Since the question specifies positive terms, we don't have alternating signs, and the convergence can only be absolute or not converge at all.
04

Conclusion

In conclusion, a series with positive terms cannot converge conditionally, as it lacks the alternating signs needed for conditional convergence. It will either converge absolutely or not converge at all.

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Most popular questions from this chapter

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)

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