2^n + n!/4^n ? so by the 'vanishing condition' as n!/4^n does not ---> 0 as n --> infinity, this part of the series diverges. however (e.g via the ratio test 2^n/4^n converges). My book concludes that due to part a, the entire series divegres. However I am struggling to see how this justifies that the whole series will diverge - I thought that if part of a series diverges and part diverges , than what the whole series does depends upon the exact convergence and divergence - i.e could diverge or converge?