Question about the comparison test

[vande060]

Homework Statement

determine whether the series diverges or converges

(∞,n=1) ∑ (n² - 1)/(3n⁴ +1)

Homework Equations

The Attempt at a Solution

(∞,n=1) ∑ (n² - 1)/(3n⁴ +1)

an = (n² - 1)/(3n⁴ +1)

i thought bn should be n²/3n⁴ = 1/3n²

lim _n-->∞ an/bn = (n² - 1)/(3n⁴ +1) * 3n²

lim _n-->∞ (3n⁴ - 3n²)/ (3n⁴ + 1)

not sure if this is going in the right direction here

 

[Mark44]

Homework Statement

determine whether the series diverges or converges

(∞,n=1) ∑ (n² - 1)/(3n⁴ +1)

Homework Equations

The Attempt at a Solution

(∞,n=1) ∑ (n² - 1)/(3n⁴ +1)

an = (n² - 1)/(3n⁴ +1)

i thought bn should be n²/3n⁴ = 1/3n²

That's a reasonable choice.

lim _n-->∞ an/bn = (n² - 1)/(3n⁴ +1) * 3n²

lim _n-->∞ (3n⁴ - 3n²)/ (3n⁴ + 1)

not sure if this is going in the right direction here

So what do you get when you take the limit?

 

[vande060]

So what do you get when you take the limit?

thats where i kept getting bogged down, let me give it another shot:

lim _n-->∞ (3n⁴ - 3n²)/ (3n⁴ + 1)

lim _n-->∞ (3 - 3/n²)/ (3 + 1/n⁴)

im _n-->∞ = 1

c>0, series converges

 

[Mark44]

Yes.