when is something absolutely convergent?

Question

For ex, would this be absolutely convergent or divergent
inf
E (-1)^(n-1) ((5n^2+2) / (4n))
n=1

Answer 1

Since you have an alternating sign function here, you don't have an absolutely convergent situation.
This is absolutely divergent, since you can take terms two at a time, and for large n, 5n^2>>2, so
each pair becomes 5 (n+1)/4 - 5n/4 or 5/4.

Answer 2

Something is said to be absolutely convergent if its absolute value is convergent. This means that there are some things which converge but their absolute value does not converge and those are the ones that we should watch out for.

The example you put up, this is not absolutely convergent. In fact, this is not conditionally convergent either because you have n^2 over n which diverges.