How can I tell if this is convergent or divergent?

enter image source here

I'm not sure what test to use. Any help is really appreciated.

1 Answer
Apr 2, 2018

It is divergent, by comparison with the harmonic series.

Explanation:

Given:

#sum_(n=1)^oo (n(n-1))/(4n^3+3n+1)#

Quick assessment

For large values of #n#, the numerator is going to be close to #n^2# and the denominator to #4n^3#, so the quotient behaves like #1/(4n)# and the sum diverges like the harmonic series #sum 1/n#.

Comparison test

We can make a comparison test with a multiple of the harmonic series by noting that:

  • The first term of the sum is #0#. Though this does not matter for us in terms of convergence or divergence, it does mean we can ignore it in our expressions below.

  • #(n-1) >= n/2# for all #n >= 2#

  • #4n^3+3n+1 < 5n^3# for all #n >= 2#

So:

#sum_(n=1)^oo (n(n-1))/(4n^3+3n+1) >= sum_(n=2)^oo (n^2)/(10n^3) >= 1/10 sum_2^oo 1/n = +oo#