need help with a math question

Question

Find the limit of the sequence, [ln(n)+(-1)ⁿ]/(3n²-1)

Would the answer of this be that the limit diverges? Or would it be zero? Would I use L'Hospital's rule while finding the limit?

Answer 1

The limit is indeed zero.

There are a few different ways that you could go about showing this result.

This is just one way to go about it (not necessarily the best one):

break the sequence into two

c_n = [ln (n) + (-1)ⁿ] / (3n² + 1) = a_n + b_n

where

a_n = ln (n) / (3n² + 1) and b_n = (-1)ⁿ / (3n² + 1)

b_n converges to 0 because the denominator goes to infinity while the numerator is finite.

You can also check that it satisfies the conditions for the convergence of an alternating sequence.

For a_n, consider the real valued function instead of the sequence and use L'Hopital's to show that the function converges.

That is, define the function f by f(x) = ln (x) / (3x² + 1) and use L'Hopital's to show that f goes to 0 as x goes to infinity, which leads to the conclusion that a_n goes to 0 as n goes to infinity.

Since a_n and b_n both go to 0 as n goes to infinity,

so does c_n