# Calc6.62

Discuss the convergence or divergence of the series $\sum _{{k=1}}^{{\infty }}{\frac {1}{\ln k}}$.
$\sum _{{k=1}}^{{\infty }}2^{k}{\frac {1}{\ln 2^{k}}}={\frac {1}{\ln 2}}\sum _{{k=1}}^{{\infty }}{\frac {2^{k}}{k}}$
converge or diverge together. It is well known that the terms of $2^{k}$ increase much more quickly than the terms of $k$ so we know that these terms do not go to 0, but instead increase to $\infty$ and thus this series diverges. We can prove this using the ratio test:
$\lim _{{k\rightarrow \infty }}\left|{\frac {{\frac {2^{{k+1}}}{k+1}}}{{\frac {2^{k}}{k}}}}\right|=\lim _{{n\rightarrow \infty }}{\frac {2k}{k+1}}=2\,$