# Calc6.55

Discuss the convergence or divergence of the series $\sum _{{k=3}}^{{\infty }}{\frac {1}{\ln(\ln k)}}\cos \left({\frac {k\pi }{3}}\right)$.

We know ${\frac {1}{\ln(\ln k)}}\rightarrow 0$ as $k\rightarrow \infty$, though at an extremely slow rate. So, if we can show that the sequence of partial sums

$\{s_{n}\}=\left\{\sum _{{k=3}}^{{n}}\cos \left({\frac {k\pi }{3}}\right)\right\}$

is bounded, then we can conclude by Dirichlet's test that our given series converges.

$\cos 0=1\,$

$\cos {\frac {\pi }{3}}={\frac {1}{2}}$

$\cos {\frac {2\pi }{3}}=-{\frac {1}{2}}$

$\cos \pi =-1\,$

$\cos {\frac {4\pi }{3}}=-{\frac {1}{2}}$

$\cos {\frac {5\pi }{3}}={\frac {1}{2}}$

And, at this point, we get repeat since $\cos(2\pi )=\cos 0$. Thus, our sequence of partial sums is (notice k starts at 3)

$\{s_{n}\}=\{-{\frac {1}{2}},-{\frac {3}{2}},-2,-{\frac {3}{2}},-{\frac {1}{2}},0,-{\frac {1}{2}},-{\frac {3}{2}},-2,-{\frac {3}{2}},-{\frac {1}{2}},0,...\}$

Thus, this sequence is bounded and, by Dirichlet's test, this series converges.