Solution
= Solution
Every term is nonnegative. Since $1-\cos x\leq x^2/2$,
$$
0\leq1-\cos(\pi/k)\leq\frac{\pi^2}{2k^2}.
$$
Comparison with $\sum k^{-2}$ shows that the partial sums are bounded. Being increasing, they converge to the finite <limit>
$$
\boxed{\sum_{k=1}^{\infty}(1-\cos(\pi/k)).}
$$