1. ## Absolute convergence problem

how would you determine if if the series converges absolutely, conditionally or diverges?

series n=1 to infinity (-5)^-n

2. Let $a_n = (-5)^{-n} = (-1)^{-n}5^{-n}$. Then $|a_n| = 5^{-n}$.

Now, notice that: $\sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n$

which is a geometric series.

3. Originally Posted by o_O
Let $a_n = (-5)^{-n} = (-1)^{-n}5^{-n}$. Then $|a_n| = 5^{-n}$.

Now, notice that: $\sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n$

which is a geometric series.
Just to add $\sum_{n=1}^\infty (-5)^{-n} = \sum_{n=1}^\infty \left(\frac{-1}{5}\right)^n$ is also geometric.