The alternating series in question is \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{5 n+1} \).
To see whether it converges or diverges, we apply the Alternating Series Test. This test requires two conditions to be met:
1) The sequence of terms must be decreasing. This means that for all \( n \), \( a_{n+1} \leq a_{n} \).
2) The limit, as \( n \) approaches infinity, of the sequence, must be 0.
Let's consider the sequence of absolute values, \( a_n = 1/(5n+1) \).
For the first condition, the terms of the sequence must be decreasing. If we compare \( a_{n+1} \) with \( a_n \), we have \( a_{n+1} = 1/(5(n+1)+1) = 1/(5n+6) \). As we increase \( n \), the denominator (5n + 6) gets larger, which makes the whole fraction smaller. Thus, \( a_{n+1} \leq a_{n} \), satisfying the first condition.
For the second condition, as \( n \) approaches infinity, the term \( a_n = 1/(5n+1) \) tends to 0, satisfying the second condition.
Since the sequence satisfies both conditions, according to the Alternating Series Test, the series converges.
But, the question also asks if it converges 'Absolutely' or 'Conditionally'.
To check for absolute convergence, we need to see whether the series given by absolute values of terms, \( \sum_{n=1}^{\infty} \frac{1}{5n+1} \), converges. Here, little analysis will show this arrangement appears somewhat like a harmonic series which is known to diverge. Hence, this series does not converge absolutely.
In conclusion, we can say that the alternating series given does not converge absolutely, but it does converge conditionally.