Not sure how to answer you. Non-convergent series are unpleasant and subtle things most 1st year mathematics text books will tell you the convergence properties of this sum. Instead let me just quote from Abel in 1828, "Divergent series are the invention of the devil and it is shamful to base on them any demonstration whatsoever". Hardy's excellent book on the subject explains the rights and wrongs; the good the bad and the ugly.