Question

Using the definition of convergence, how do you prove that the sequence `(-1)^n/(n^3-ln(n))` converges from n=1 to infinity?

Answer 1

Take `N>(2/epsilon)^(1/3)`

Explanation:

The limit is 0 so

`abs(s_n-L)=1/(n^3-lnn)< 1/(n^3-n)=1/(n(n^2-1))<1/(n*n^2/2)=2/n^3`

Now for each `n>N>(2/epsilon)^(1/3)` we get

`abs(s_n-L)<2/n^3<2/((2/epsilon)^(1/3))^3=epsilon`