Answer
The series is divergent and $b_n$ is not decreasing so, we cannot apply the Alternating Series test.
Work Step by Step
We are given that $\Sigma_{n=1}^\infty (-1)^{n+1} b_n$
For odd values, we have $b_n=\dfrac{1}{n}$
For even values, we have $b_n=\dfrac{-1}{n^2}$
It has been seen that $\dfrac{-1}{n^2}\lt \dfrac{-1}{n+1}$ (For every $\dfrac{1}{n^2}$ term, which is less than the absolute value).
Therefore, $b_n$ is not decreasing and thus AST can not be applied.
Further, $\Sigma_{n=1}^\infty (-1)^{n+1} b_n=\Sigma_{n=1}^\infty \dfrac{1}{2n-1}+\Sigma_{n=1}^\infty\dfrac{-1}{(2n)^2}$
But $\Sigma_{n=1}^\infty \dfrac{1}{2n-1} \gt \Sigma_{n=1}^\infty\dfrac{-1}{(2n)^2}$
The series $\Sigma_{n=1}^\infty \dfrac{1}{2n-1}$ will diverge by the comparison test.
Hence, it has been verified that the series is divergent and $b_n$ is not decreasing so, we cannot apply the Alternating Series test.
