Use the comparison test to determine whether the series converges or diverges.
n < en for n ≥ 1
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 < en to replace en with n:
Then make the numerator bigger too. Since cos n is always between -1 and 1,
2 – cos n ≤ 3.
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,