# Thread: Conditional and absolute convergence

1. ## Conditional and absolute convergence

For each $\displaystyle y \in (0,1]$, find a series of the form $\displaystyle \sum\limits_{n=1}^{\infty}{a_nn^{-x}}$ which is absolutely convergent for $\displaystyle x>1$ but not for $\displaystyle x<1$ and is (conditionally) convergent for $\displaystyle x>y$ but not for $\displaystyle x<y$.

2. Originally Posted by Newtonian
For each $\displaystyle y \in (0,1]$, find a series of the form $\displaystyle \sum\limits_{n=1}^{\infty}{a_nn^{-x}}$ which is absolutely convergent for $\displaystyle x>1$ but not for $\displaystyle x<1$ and is (conditionally) convergent for $\displaystyle x>y$ but not for $\displaystyle x<y$.
Is the sequence supposed to depend on $\displaystyle y$?

3. Yes; we may assume we're given a $\displaystyle y \in (0,1]$ and we want to find a series as stated (by defining $\displaystyle a_n$, which are allowed to depend on the given $\displaystyle y$).