St. Paul challenges the Corinthians, in preparation for his visit, to examine [themselves] to see whether [they] are in the faith (v. 5). The kind of testing Paul envisions is that which proves the worth or genuineness of something and in this case, it is the Corinthians' faith that is to be proven. The Corinthians have professed a belief in Christ, but does their life match their profession? If the life of the congregation is not in conformity with the truths of the gospel, it negates any claim to standing firm in the faith (1 Cor 16:13).
Of course, such testing requires that the Corinthians possess the ability to recognize Christ's presence among them. Yet if the Corinthians have their spiritual wits about them, they will discover that Paul "is not counterfeit" or have not failed the test. The idea is that as they examine themselves and find themselves to be genuine Christians, they will be led, in turn, to evaluate Paul and see that he also has passed the test. Paul reminded the church that the work at Corinth was and continues to be a team effort. So to question Paul's authenticity [genuineness] is to question the authenticity of the team effort (that is, Paul, Silas, Timothy and Titus). In the final analysis, however, what matters to Paul is that they will do what is right--even if it means that his work at Corinth may seem to have failed (v. 7). Not that Paul expected to fail the test. But such a price would be worth paying if it guaranteed that the Corinthians would do the right thing. Just as Christ's sacrificial love compels Paul to preach the gospel (5:14), so too his commissioning as Christ's apostle hems him in to doing only what advances the gospel.
What better way to end the letter than by pointing to the perfect model of "congregational" unity--the unity of Father, Son and Holy Spirit [Doctrine of The Holy Trinity]! So Paul's concluding benediction is more than a theological fanfare. The order "Christ's grace," "God's love" and "the Holy Spirit's fellowship" is very practical. Through Christ's gift of Himself we experience, in the most concrete terms, God The Father's love for us and the Spirit's power to fashion us into a oneness that serves as a beacon of hope in a fragmented and broken world.