If you have a geometric alternating series, and you prove that the series is converging by doing geometric series test, and NOT alternating series test, then does that allow you to say that the series converges ABSOLUTELY?
Or should you do alternate series test also to say that it converges absolutely? I was confused because I know that ratio test and root test allows you to say that it converges absolutely without doing the alternate integral test.
What about the other tests like comparison tests and integral test?
Thanks!
If it is a geo alternating, then proof of the geometric convergence test is sufficent for absolute convergence.
Any test, and there are several, that says the series converges absolutely is sufficent: others are not required.
If you have a geometric alternating series and you prove that the series is converging using the geometric series test, you can indeed say that the series converges absolutely. The alternating series test is not necessary in this case.
The geometric series test checks for convergence of a series by looking at the common ratio between consecutive terms. If the absolute value of the common ratio is less than 1, the series converges. In the case of a geometric alternating series, the alternating signs of the terms do not affect the convergence behavior, as the series still follows a geometric pattern.
On the other hand, the alternating series test specifically applies to alternating series where the terms alternate in sign. It checks whether the terms in the series decrease in magnitude as you progress further in the series. If this condition is met, the alternating series converges.
It's worth noting that other tests, such as the ratio test or the root test, can also be used to determine the absolute convergence of a series without relying on the alternating series test. These tests analyze the behavior of the terms in relation to their indices or exponents to determine if the series converges absolutely.
Additionally, there are other tests like the comparison tests and the integral test that can be used to determine convergence of a series. These tests compare the given series to known convergent or divergent series or analyze the behavior of functions through integration, respectively.
In summary, for a geometric alternating series, proving convergence using the geometric series test is sufficient to conclude absolute convergence. However, depending on the specific series, other tests may also be applicable and provide different ways to determine absolute convergence.