Calc6.41

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n}}.

This is another p-series, this time with p=1, so we already know that this series diverges. However, let us look at the ratio test once more.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {1}{n+1}}}{{\frac  {1}{n}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {n}{n+1}}\right|=1\,

Again, the ratio test is inconclusive for a p-series. In fact, the ratio test is inconclusive for any p-series. The previous example showed a p-series which converged and this shows a p-series which diverges. The ratio test is inconclusive in both cases. The integral test works well for p-series but the ratio test does not.


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