does this series converge, and if so is it absolutely convergent?

the series from n=1 to infinity of
((-1)^*n+1))/n^4

I found that by the ratio test it was inconclusive, so no abs. conv
is this right? and how do i know if it is simply convergent?

I still don't understand what your ^* represents

To determine if the series from n=1 to infinity of ((-1)^(n+1))/n^4 converges, we can first check if it converges absolutely. If it does not converge absolutely, we can then check if it converges conditionally.

To investigate absolute convergence, we need to examine the series you provided: ((-1)^(n+1))/n^4. To determine if this series converges absolutely, we look at the absolute values of the terms. The absolute value of ((-1)^(n+1))/n^4 is just 1/n^4.

Let's use the p-series test to determine if the series 1/n^4 converges. The p-series test states that a series of the form 1/n^p converges if p > 1, and diverges if p <= 1.

In our case, p = 4, which is greater than 1. So the series 1/n^4 converges. Since the series ((-1)^(n+1))/n^4 has the same terms as 1/n^4 in absolute value, it also converges absolutely.

Therefore, your conclusion that the series converges but does not converge absolutely is not correct. The series from n=1 to infinity of ((-1)^(n+1))/n^4 actually converges absolutely.

To determine if the series converges conditionally, we need to consider whether the series converges when the signs of the terms alternate. In this case, the signs do alternate since we have (-1)^(n+1) in the numerator. Therefore, the series also converges conditionally.

In summary:
- The series from n=1 to infinity of ((-1)^(n+1))/n^4 converges absolutely.
- The series from n=1 to infinity of ((-1)^(n+1))/n^4 also converges conditionally.

If the ratio test is inconclusive, it is often helpful to try other convergence tests, such as the comparison test, limit comparison test, or integral test, to determine the convergence or divergence of a series.