Question

Determine whether the following series converges or diverges: sum_(n = 1)

Series
ANSWERED
asked 2021-08-19
Determine whether the following series converges or diverges:
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{n}^{{3}}+{n}^{{4}}}}{{n}^{{5}}}.\)

Expert Answers (1)

2021-08-20
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{{n}^{{3}}+{n}^{{4}}}}{{n}^{{5}}}=\)
\(\displaystyle={\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}^{{2}}}+{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}}\)
As \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}^{{2}}}\) is convergent by p-series test and \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}}\) is divergent,
Series diverges because we can break the series up into a convergent p-series and a divergent p-series.
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