Problem 7
Question
Is it possible for a series of positive terms to converge conditionally? Explain.
Step-by-Step Solution
Verified Answer
Answer: No, a series of positive terms cannot converge conditionally. If a series of positive terms converges, it will always converge absolutely.
1Step 1: Understanding conditional and absolute convergence
Conditional convergence means that a series converges, but it does not converge absolutely. Absolute convergence is when the series of the absolute values of the terms converges. If a series of positive terms converges conditionally, it means that the series converges, but its absolute values—which are just the terms themselves, since they're positive—do not converge.
2Step 2: Evaluating convergence of series of positive terms
In a series with positive terms, the sum always increases monotonically, i.e., the sum of the terms in the series will continue to grow as more terms are added. If the sum of such a series converges, that means it converges absolutely.
3Step 3: Drawing the conclusion
A series of positive terms will either converge absolutely or diverge, since the sum of the positive terms continues to grow. This means that a series of positive terms cannot converge conditionally, as a convergent series of positive terms will also concurrently converge absolutely.
Other exercises in this chapter
Problem 6
Explain how the methods used to find the limit of a function as \(x \rightarrow \infty\) are used to find the limit of a sequence.
View solution Problem 6
Given the series \(\sum_{k=1}^{\infty} k\), evaluate the first four terms of its sequence of partial sums \(S_{n}=\sum_{k=1}^{n} k\)
View solution Problem 7
Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.
View solution Problem 7
Evaluate each geometric sum. $$\sum_{k=0}^{8} 3^{k}$$
View solution