tn The infinitive περιπατῆσαι (peripatēsai, “to walk, to live, to live one’s life”) is best taken as an infinitive of purpose related to “praying” (προσευχόμενοι, proseuchomenoi) and “asking” (αἰτούμενοι, aitoumenoi) in v. 9 and is thus translated as “that you may live.”
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tnBDAG129s.v.ἀρεσκεία states that ἀρεσκείαν (areskeian) refers to a “desire to pleaseεἰς πᾶσαν ἀ. to please (the Lord) in all respectsCol 1:10.”