Question

For what integer k, k>1, will both sigma infinity when n=1 ((-1)^(kn))/n and sigma infinity when n=1 (k/4)^n converge?

Answer 1

Have you tried the alternating series test for the first series?

Answer 2

\[\large\sum_{n=1}^\infty\frac{(-1)^{kn}}{n}\text{ will converge if}\\
\lim_{n\to\infty}\frac{1}{n}=0, \text{ and}\\
\left\{\frac{1}{n}\right\}\text{ is a decreasing sequence.}\]
However, this series will only converge if k is odd (1, 3, 5, ...). Otherwise, you have the series
\[\sum_{n=1}^\infty\frac{1}{n}, \text{ which diverges.}\]

Answer 3

The second series is a geometric series. What do you know about the convergence conditions of a geo-series?