Problem 108 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.