Conditional and Absolute Convergence

Question

Test to see if the following functions converges absolutely or conditionally $\sum\frac{(-1)^k}{{(k^5-4)^{1/5}}}$. For the absolute convergence I understand that I have to consider the function $\frac{1} {(k^5-4)^{1/5}}$ and test it for convergence. for conditional convergence I guess I could use the alternating series test or the ratio test but i'm not sure how! help please! :D

Answer 1

To test for absolute convergence, note that the $k$-th term, at least from $k=2$ on, has absolute value greater than $\dfrac{1}{k}$.

For conditional convergence, use the Alternating Series Test (from $k=2$ on). In addition to the obvious sign stuff, we need to check that the terms decrease in absolute value, and approach $0$. This should not be hard.

Remark: For absolute convergence, the Ratio Test is in this case inconclusive. For conditional convergence, it is irrelevant.