Use the comparison test to determine whether the series converges or diverges.

Answer

This kind-of-sort-of looks like , which converges, guess the series converges. We need to find a convergent series with bigger terms. To make a fraction bigger we can make the denominator smaller and/or make the numerator bigger.

To make the denominator smaller, use the fact that *n* < *e*^{n} to replace *e*^{n} with *n*:

Then make the numerator bigger too. Since cos n is always between -1 and 1,

2 – cos *n* ≤ 3.

This means

We know that

converges because of the p-test and because multiplication by a constant doesn't affect whether the series diverges. This means the series with smaller terms,

has to converge.