Question

How can I tell if this is convergent or divergent?

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

Answer 1

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`