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Section 10.7 : Comparison Test/Limit Comparison Test

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

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