For the first serie, i would prove that it converges absoloutly using that
edit: my idea doesn't work's
For the second, mmmm, let me think xd
Notice that for both of these series that as n (or k) gets very large, the value 1/n (or 1/(4k)) gets very close to zero. Consider that the derivative of sin(x) and tan(x) at points very near zero is nearly equal to one, which is like saying at points very near zero these functions behave similarly to f(x)=x. So, the functions you give could be said to be comparable to f(x)=1/x.
Brian