sequences and series - Testing for absolute convergence? $frac{(-1)^n}{5n+1}$

I'm trying to test the summation $\sum^\infty_{n=0}\frac{(-1)^n}{5n+1}$ for absolute convergence.

By alternating series test, I can tell is is at least conditionally convergent.

However, when I used the ratio test, I got 1, which means it doesn't tell us anything.

A google search showed an answer on yahoo answers using the limit comparison test, using the harmonic series to compare it with, but they seemed to ignore the $(-1)^n$...

The answer in the book says it is conditionally convergent, but I can't work out how to show that it is not absolutely convergent.

Any ideas?