I was wondering if I could get a hint, and a hidden answer on these two series. We are suppose to find out if they converge absolutely, or conditionally. I am stuck on the test I should use.
(1) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}k^k}{(k+1)^{k+1}}$. Which I think that I should use the comparison test on.
(2) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} k^k}{(k+1)^k}.$ Which I think will be similar to the first.
$1)$ The series is conditionally convergent .
Apply alternating series test ( we can prove that it satisfies the conditions using the logrithm as mentioned by @Tim ) to prove it converges. Then using comparison test (with ${1\over 2k}$ ), we can show that it doesn't converge absolutely.
$2)$ The series is not absolutely convergent because the limit of the terms is not zero. Since the terms don't go to zero, we cannot have convergence ( this is discussed in many questions regarding the converse of the alternating series test ) . Therefore , theis series diverges.