# Calc6.63

Discuss the convergence or divergence of the series $\sum _{{k=1}}^{{\infty }}{\frac {1}{k^{2}}}$.
$\sum _{{k=1}}^{{\infty }}2^{k}{\frac {1}{(2^{k})^{2}}}$
$\sum _{{k=1}}^{{\infty }}2^{k}{\frac {1}{(2^{k})^{2}}}=\sum _{{k=1}}^{{\infty }}{\frac {1}{2^{k}}}$
This is just a geometric series with common ratio ${\frac {1}{2}}$ and first term ${\frac {1}{2}}$ which is well known to converge to 1. Thus, this series converges and therefore the original series converges by the Cauchy condensation test.