てご覧（らん） is a 尊敬語（そんけいご） (honorific language) expression in Japanese that is used in a similar way to
なさい. It is usually translated as ‘please try to (A)’, where (A) is the verb that ご覧（らん） is connected to, through the use of the 接続助詞（せつぞくじょし） て.
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この本（ほん）を読（よ）んでごらん。この本（ほん）を読（よ）んだら日本（にほん）について色々（いろいろ）と学（まな）べるよ。
Try reading this book. If you read it, you will learn various things about Japan.
外（そと）を見（み）てごらん。虹（にじ）が出（で）ているよ。
Look outside. There is a rainbow.
However, despite being translated as ‘please try to (A)’, the literal meaning of 覧（らん） is ‘to look’, or ‘to peruse’ (to look at one’s own leisure, without needing to rush). Due to this, てごらん can be thought of as being quite similar to
てみる ‘to try’, but with the added emphasis of ‘try as you like, when it suits you’.
As ご覧（らん） is 尊敬語（そんけいご）, it cannot be used to refer to your own actions, or things that you will ‘try’. It will only ever be used to make suggestions to other people about what they should try.
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これ食（た）べてごらん。あなただったら絶対（ぜったい）に好（す）きだと思（おも）うよ。
Try eating this. If it’s you, I think you’ll definitely like it.
Sometimes ごらん and
なさい will be paired together. This slightly increases the strength of the suggestion, and may be perceived as a very light command. Because of this, ごらんなさい is primarily used by superiors/elders, when requesting that a subordinate/younger person try something.