First, find the general form of your series.
So we know that
By the comparison theorem, we know that the first series converges if the second series converges. By the theorems about summation by parts, we know that the second series converges (that is quite a long proof which is why i do not write it down, should be in every introductury analysis book, however). You can derive your values by setting n = k + 5 where 5 are the values you already have given and k is your row in the table.
If I remember it correctly the result of this series is but it is not that easy to show it.