Divergent Alternating Series

Question

Need help in finding an alternating series:

S = $\sum_{n=1}^{\infty}(-1)^{n+1}b_n$

where

$\lim_{n\to \infty}b_n = 0$

$b_n > 0$ but only $\forall n \ge 1$

such that S diverges

Answer 1

Try something like $b_{2n}=\dfrac{1}{2n}$ and $b_{2n+1}=\dfrac{1}{(2n+1)^2}$

Answer 2

$$ b_n=\frac{1}{(n+1)^p+(-1)^n},\quad 0<p\le\frac12. $$

Answer 3

$(-1)^n \frac{n!!}{(n+1)!!}$ where

n!!= n(n-2)(n-4)...2 if n is even

n!!= n(n-2)(n-4)...3 if n is odd