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Section 10.6 : Integral Test

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

1. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{4}{{{{\left( {\sqrt n } \right)}^3}}}} \)
2. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{1}{{\sqrt[7]{{{n^2}}}\,\,\sqrt[6]{n}}}} \)
3. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{2n + 1}}} \)
4. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{8}{{{{\left( {n + 10} \right)}^2}}}} \)
5. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^2} + 1}}} \)
6. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{\ln \left( n \right)}}{n}} \)
7. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{n^3}}}{{{n^4} + 1}}} \)
8. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{n^3}}}{{{{\left( {{n^4} + 1} \right)}^2}}}} \)
9. \( \displaystyle \sum\limits_{n = 4}^\infty {\frac{4}{{{n^2} - n - 6}}} \)
10. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{9}{{{n^2} + 5n + 4}}} \)
11. \( \displaystyle \sum\limits_{n = 0}^\infty {n\,{{\bf{e}}^{ - n}}} \)