Hi,

I have have to determine if the following series is convergent or divergent.

$\displaystyle \displaystyle \sum_{k= 1}^{\infty} \frac{(k!)^{\frac{1}{k}}}{k}}= \sum_{k= 1}^{\infty} \root k \of {k!}{\frac{1}{k}$

$\displaystyle \displaystyle a_k=\frac{(k!)^{\frac{1}{k}}}{k}}$

$\displaystyle \displaystyle b_k=\frac{1}{k}}}$

$\displaystyle b_k$ is a p-series and diverges.

$\displaystyle \displaystyle\lim_{k \to \infty} \frac{a_k}{b_k}}=\root k \of {k!}=1$

According to direct comparison rule, since $\displaystyle b_k$ is divergent then the given series must also be divergent.(1)

Is (1)correct?

I am also unsure about the following limit:

(2) $\displaystyle \displaystyle\lim_{k \to \infty} \sqrt{(k!)}=1$

Is (2) correct?

I would appreciate a response.

Thank you