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Section 10.10 : Ratio Test

For each of the following series determine if the series converges or diverges.

1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{n^3} + {n^2}}}{{\left( {n + 1} \right)!}}} \)
2. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{n + 2}}{{{5^{1 - n}}\left( {n + 1} \right)}}} \)
3. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{\left( {2n - 1} \right)!}}{{\left( {3n} \right)!}}} \)
4. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 2} \right)}^{4 + n}}}}{{3{n^2} + 1}}} \)
5. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{4^{1 + \frac{1}{2}n}}{n^2}}}{{{3^{2 + n}}\left( {n + 3} \right)}}} \)
6. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{4}{{{{\left( { - 1} \right)}^{n + 2}}\left( {{n^2} + n + 1} \right)}}} \)
7. \( \displaystyle \sum\limits_{n = 3}^\infty {\frac{{{6^{ - 2n}}\left( {n - 4} \right)}}{{{4^{3 - 2n}}\left( {2 - {n^2}} \right)}}} \)
8. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^n}\left( {n + 1} \right)}}{{{n^2} + 1}}} \)