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Chapter 7: Problem 108

Using the Ratio Test, it is determined that an alternating series converges. Does the series converge conditionally or absolutely? Explain.

Short Answer

Expert verified
The series converges absolutely. This is because the Ratio Test, which verifies absolute convergence, determined the series to be convergent.

Step by step solution

01

Understanding ratio test and absolute convergence

Firstly, it is important to note that the Ratio Test is normally used to verify absolute convergence rather than convergence of an alternating series. If the Ratio Test proves a series to be convergent, it affirms that that series is absolutely convergent.
02

Infer the conclusion from Ratio Test

Since the Ratio Test was applied and it determined the series to be convergent, it can thus be inferred that the series is absolutely convergent.
03

Final conclusion regarding convergence

In conclusion, the series doesn't just converge, but it converges absolutely according to the given condition that it passed the Ratio Test.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(n>1\), then \(n !=n(n-1) !\)

Prove that if \(\left\\{s_{n}\right\\}\) converges to \(L\) and \(L>0,\) then there exists a number \(N\) such that \(s_{n}>0\) for \(n>N\).

Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\\{\sqrt[n]{n}\\} .\) If the sequence converges, find its limit.

Prove, using the definition of the limit of a sequence, that \(\lim _{n \rightarrow \infty} \frac{1}{n^{3}}=0\)

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$
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