inf (x ^ 2) + 1

sum -----------------

x=1 3 x ^ 3 - x + 2

I find potentially convergent

If you meant to write the series , then you have a divergent series using, for example, the comparison test: , and since the last one is the general term of a multiple of the harmonic series' general term then it diverges. Tonio
Then I say discard all but x ^2 / x^3, so that simplifies to be:

1/x

use p series test ast 1/x = 1/x^1, where the 1 is <=1 and hence divergent.

x ^ 2 + 1

---------------

3 x^ 2 -x + 2

-----------------

1/x

= ... lim

x->inf x(x^2+1)

----------

3x^2-x+2

= inf which is > 1

so the answer is... this is divergent by limit comparison test.

Is that right?