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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}+4}}.

The series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}}} converges because it is a p-series with p=3>1 and our series looks a lot like this series so we would think that is also converges. Thankfully we have the direct comparison test so that we can test our series with other series with known convergence or divergence.

0<{\frac  {1}{n^{3}+4}}<{\frac  {1}{n^{3}}} for all n>0

We have a known convergent series with positive terms, \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}}}, and its terms are all larger, term by term, than our series which also has positive terms. Thus, by the direct comparison test, our series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}+4}} converges. This makes complete sense. If we have an infinite sum of positive numbers which converges to a finite number, then it makes perfect sense that another infinite sum of smaller positive numbers would not only converge, but converge to a smaller positive number.

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