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?

Is ((-1)^*n+1)) supposed to mean

[(-1)^(n+1)] ?

If so, an infinite series of positive 1/n^4 terms is convergent, based on the integral test. A similar series with alternating + or - terms of the same magnitude must therefore also converge.

To determine if the series converges, we can start by checking if it is absolutely convergent.

Absolute convergence means that the series converges when we consider the absolute values of each term. In other words, we take the series:

∑(|((-1)^(n+1))/n^4|)

In this case, each term is positive because we take the absolute value. Therefore, we can simplify it as follows:

∑(1/n^4)

To check if this series converges, we can use the p-series test. For any positive real number p, the p-series ∑(1/n^p) converges if p > 1 and diverges if p ≤ 1.

In our case, we have p = 4, which is greater than 1. So, the series ∑(1/n^4) converges. Consequently, the original series ∑(((-1)^(n+1))/n^4) is absolutely convergent.

Now, to determine if the series is simply convergent (without considering absolute values), we can use the alternating series test. For an alternating series ∑((-1)^(n+1)) * b_n, where b_n > 0, the series converges if all of the following conditions are met:

1. The terms b_n decrease in magnitude (|b_n+1| ≤ |b_n|).
2. The limit of b_n as n approaches infinity is 0.
3. The terms b_n eventually approach 0.

In our series, b_n = 1/n^4. Let's check these conditions:

1. The terms 1/n^4 decrease in magnitude as n increases, so the first condition is satisfied.
2. The limit of 1/n^4 as n approaches infinity is 0, so the second condition is satisfied.
3. The terms 1/n^4 eventually approach 0 as n increases, so the third condition is satisfied.

As all conditions of the alternating series test are fulfilled, the series is simply convergent.

In summary, the given series ∑(((-1)^(n+1))/n^4) is absolutely convergent since the series ∑(1/n^4) converges. Additionally, the series is simply convergent since it satisfies the conditions of the alternating series test.