Question

Does the series (cos(k/3))^2/2^(k/3)+1 from k=1 to infinity converge?

Answer 1

since cos(u) < 1,

your terms are less than 1/(2^(k/3)+1)

sum 1/2^(k/3) converges, and your terms are less than 1/2^(k+3), your series converges