# Calc6.51

Discuss the convergence or divergence of the series $\sum _{{k=1}}^{{\infty }}{\frac {\sin {\frac {k\pi }{2}}}{k}}$.
To show convergence for this series we will use Dirichlet’s test. The terms of this sequence are the product of ${\frac {1}{k}}$ and $\sin {\frac {k\pi }{2}}$. The first sequence of numbers decreases toward 0 as $k\rightarrow \infty$. For the second sequence of numbers, $\sin {\frac {k\pi }{2}}$, let us look at the partial sums
$s_{n}=\sum _{{k=1}}^{{n}}\sin {\frac {k\pi }{2}}\,$