Will Jagy: Sorry for pinging you on an unrelated post - I've tried this way since I cannot comment under a deleted post. I have seen your answer to The asymptotic behavior of $n\ln n -n$ where you included a picture which might be useful for users trying to solve this problem. Since the question is being repeatedly undeleted and deleted, it seems unlikely that it will survive on the site. — Martin Sleziak 1 min ago

There are several posts on this site which are about very similar limit. For example, the question Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. does not have "integral vs. sum" answer. So I thought that perhaps by posting your answer there, it would be more likely that it remains on the site. Feel free to ping me here or in chat when these comments are no longer needed and I can delete them. — Martin Sleziak 10 secs ago

@MartinSleziak thanks, I moved it to math.stackexchange.com/questions/376988/… as you suggested — Will Jagy 25 mins ago