How to find a sequence $\{b_n\}^{\infty}_{n=1}$based on the following assumption? How to find a sequence $\{b_n\}^{\infty}_{n=1}$ such that 
$b_n\gt 0$ for all $n\ge 1$ and 
$\lim\limits_{x\to\infty}b_n=0$ and 
the series $\sum_1^\infty (-1)^{n+1}b_n$ is divergent.
For alternating series, the sequence converges if and only if $b_n\to 0$ based on the assumption that $b_n\gt 0$ for all $n\ge 1$, $\lim\limits_{x\to\infty}b_n=0$ and the series is decreasing. So I think we have to find a nondecreasing sequence. But how do you find a nondecreasing function that has $\lim\limits_{x\to\infty}b_n=0$?

Hint: Interleave the "bad" sequence $1,1/2,1/3,1/4,1/5,\dots$ with the good sequence $1,1/2,1/4,1/8,1/16,\dots$.
If a formula is desired, if $n=2k-1$ we let $b_n=\frac{1}{k}$, and if $n=2k$ we let $b_n=\frac{1}{2^{k-1}}$. 
