N3
てごらん
(Please) try to, (Please) look
外を見てごらん。虹が出ているよ。
Look outside. There is a rainbow.
Explanation
「~てごらん」 is a gentle way of saying 「~てみなさい」: 'try it', 'have a look'. Parents to children, teachers to students and elders to younger people use it to encourage kindly: 「これ、食べてごらん。すごく美味しいよ」 (Try this. It's really good), 「空を見てごらん」 (Look at the sky). The full form is 「~てごらんなさい」.
Mixed up with
Use it only downward. Never to elders, superiors or people you don't know.
「~てみて(ください)」 works for anyone: friends, or strangers with 「ください」.
「~てみなさい」 is stricter and sounds like instruction, less warm than 「~てごらん」.