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?
Best Answer
The absolute series would be: $\sum_{n=0}^\infty\frac{1}{5n+1}\geq\sum_{n=0}^\infty\frac{1}{6n}$ (at least for sufficiantly large $n$ it is). Then:$$\sum_{n=0}^\infty\frac{1}{6n}=\frac16\sum_{n=0}^\infty\frac{1}{n}$$ Wich is obviously divirgent.