Using the definition of convergence, how do you prove that the sequence (-1)^n/(n^3-ln(n))(1)nn3ln(n) converges from n=1 to infinity?

1 Answer
Nov 10, 2016

Take N>(2/epsilon)^(1/3)N>(2ε)13

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|snL|=1n3lnn<1n3n=1n(n21)<1nn22=2n3

Now for each n>N>(2/epsilon)^(1/3)n>N>(2ε)13 we get

abs(s_n-L)<2/n^3<2/((2/epsilon)^(1/3))^3=epsilon|snL|<2n3<2((2ε)13)3=ε