are the following sums,with general tern an, convergent or divergent.

an=(1+n)/n^2

my book says convergent but i think divergent since

an=1/n^2 + 1/n >1/n

2)

an=(1+n)/(2+n^2)

book says convergent but i think divergent

this time i think an>(1+n)/(n+n^2)=1/n

3) an=1/[n ln(n)] convergent

compare with un=1/n^2 so an/un=n/ln(n) which tends to 1

4) an=ln(n)/n^2..

i think convergent but not sure how to show this.

5) an=ln (n)/(2n^3-1)

as for 4. think convergent but not doing correct comparison.

5) an=(2n-1)/n(n+1)(n+2)

this converges, i put into partial fractions and got an expression for the sum. Is there a nice comparison to show it quicker?