The words "for the obedience of faith to all the nations" recall the words of the introduction (1:5). This phrase designates God's purpose or goal in making known the gospel. Gentiles participate in the Abrahamic blessing through the obedience that flows from faith. Paul never conceived of salvation taking root among the nations without a change of behavior. The gospel that takes hold of human beings changes them so that they become servants of righteousness. Such new behavior, however, has its roots in faith, in trusting God for strength and power to live a new life. The gospel does not summon people to exercise their own moral virtue. It calls them to put their trust in God, who raised Jesus from the dead. By trusting him they will be filled with the power to live fruitful lives.
I should mention however, that Schreiner points out that this was not the ultimate goal of Paul's life and letter, just the penultimate goal. The ultimate goal is God's glory. This goal is also found in both the prologue and the benediction:
to bring about ... for the sake of his [Christ] name ... (Rom 1:5)
Now to him who ... to the only wise God be glory forevermore through Jesus Christ. (Rom 16:26-27)
Paul is and always will be, Christ-centered.