Determine whether this series is a convergent or divergent.

Question

Determine whether this series is a convergent or divergent.

∑ _n=1^∞ 1 /n + n cos²n

Would you please explain in steps 😊

Answer 1

The series does not pass the divergence test: Limit as n goes to infinity of a_n must go to 0 for convergent series.

The first term does qualify(although it is the harmonic series and you can show divergence by the integral test: ∫ ₁^∞ 1/x dx diverges) , but the second term grows with n and cos²(n) has a value between 0 and 1. Even if the first series converged (which it doesn't) you are adding a divergent series to it. Another way to say it is that a_n doesn't have a limit at all because the cos function does not approach a limit. Even if the 1st series converged and the second term was just cos²n, the series would still diverge!