Sentral said:I know that the basic ones such as ratio, divergence, alternating series, comparison, and integral don't work with this series. I believe it uses a test that I haven't learned about, so I was wondering what that could be.
To show that a series diverges, you can use one of several tests such as the Divergence Test, the Integral Test, or the Comparison Test. These tests involve examining the behavior of the terms in the series and determining if they approach infinity or if they do not converge to a finite limit.
Yes, a series can still diverge even if its terms approach zero. This is because the terms may decrease at a slower rate than the rate at which they approach zero, resulting in an infinite sum. An example of this is the Harmonic Series, which has terms that approach zero but the series still diverges.
The Divergence Test is a basic test for divergence that states that if the limit of the terms in a series does not approach zero, then the series must diverge. This is because if the terms do not approach zero, then they cannot converge to a finite sum. However, if the limit of the terms is zero, the test is inconclusive and other tests must be used.
A series is absolutely convergent if the sum of the absolute values of its terms is finite. A series is conditionally convergent if it is convergent, but not absolutely convergent. This means that the series converges, but if the signs of the terms were changed, the series would diverge. An example of a conditionally convergent series is the Alternating Harmonic Series.
Yes, a series can still diverge even if its terms alternate between positive and negative values. This is because the terms may not approach zero quickly enough, resulting in an infinite sum. An example of this is the Alternating Harmonic Series, where the terms alternate between positive and negative but the series still diverges.