Sense

1. (with genitive, often with the associated idea of a body of water), on the other side, across
2. (absolute) on the other side
3. (with verbs of motion, followed by εἰς ‎(eis)) over or across to
4. (with the article)

Examples

• καὶ παρεγένοντο εἰς ἅλωνα Ἀτάδ, ὅ ἐστι πέραν τοῦ Ἰορδάνου, καὶ ἐκόψαντο αὐτὸν κοπετὸν μέγαν καὶ ἰσχυρὸν σφόδρα. καὶ ἐποίησε τὸ πένθος τῷ πατρὶ αὐτοῦ ἑπτὰ ἡμέρας. (Septuagint, Liber Genesis 50:10)
• καὶ εἶδον οἱ κάτοικοι τῆς γῆς Χαναὰν τὸ πένθος ἐπὶ ἅλωνι Ἀτὰδ καὶ εἶπαν. πένθος μέγα τοῦτό ἐστι τοῖς Αἰγυπτίοις. διὰ τοῦτο ἐκάλεσε τὸ ὄνομα αὐτοῦ Πένθος Αἰγύπτου, ὅ ἐστι πέραν τοῦ Ἰορδάνου. (Septuagint, Liber Genesis 50:11)
• καὶ ἐξάραντες ἐξ Ὠβώθ, καὶ παρενέβαλον ἐν Ἀχαλγαὶ ἐκ τοῦ πέραν ἐν τῇ ἐρήμῳ, ἥ ἐστι κατὰ πρόσωπον Μωάβ, κατ ἀνατολὰς ἡλίου. (Septuagint, Liber Numeri 21:11)
• καὶ ἐκεῖθεν ἀπάραντες παρενέβαλον εἰς τὸ πέραν Ἀρνῶν ἐν τῇ ἐρήμῳ, τὸ ἐξέχον ἀπὸ τῶν ὁρίων τῶν Ἀμορραίων. ἔστι γὰρ Ἀρνῶν ὅρια Μωὰβ ἀνὰ μέσον Μωὰβ καὶ ἀνὰ μέσον τοῦ Ἀμορραίου. (Septuagint, Liber Numeri 21:13)
• Καὶ εἶπε Κύριος πρὸς Μωυσῆν. ἀνάβηθι εἰς τὸ ὄρος τὸ ἐν τῷ πέραν τοῦ Ἰορδάνου καὶ ἰδὲ τὴν γῆν Χαναάν, ἣν ἐγὼ δίδωμι τοῖς υἱοῖς Ἰσραὴλ ἐν κατασχέσει. (Septuagint, Liber Numeri 27:12)

Source: Ancient Greek entries from Wiktionary