1. ## is this convergent?

inf (x ^ 2) + 1
sum -----------------
x=1 3 x ^ 3 - x + 2

I find potentially convergent

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?

2. Originally Posted by orendacl
inf (x ^ 2) + 1
sum -----------------
x=1 3 x ^ 3 - x + 2

I find potentially convergent

If you meant to write the series $\displaystyle \sum^\infty_{n=1}\frac{n^2+1}{3n^3-n+2}$ , then you have a divergent series using, for example, the comparison test: $\displaystyle \frac{n^2+1}{3n^3-n+2}\geq \frac{n^2}{4n^3}=\frac{1}{4n}$ , 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?
.