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"entertain you" is correct and usable in written English.
It is typically used as an imperative, as in the following example: "Let us entertain you with a song!".
Exact(60)
A good entertaining website that cures boredom is Bored.com - It has many games that entertain you.
They'll entertain you".
Let them entertain you.
I want to entertain you".
"We want to entertain you.
"Let me entertain you," the actor says.
"You come, and they entertain you".
"Let me entertain you," he warbled.
"I wasn't trying to entertain you.
They are here to entertain you.
I am not here just to entertain you.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com